Talk:Zermelo–Fraenkel set theory/Archive 1

The is the archive file "Talk:Zermelo–Fraenkel set theory/Archive 1".

Some early questions

I'm confused: are the listed axioms for ZFC or von Neumann thingy lala? -Martin

They're ZFC; I've clarified. As a separate point, isn't the empty set axiom redundant here? It seems to follow from infinity and replacement. Matthew Woodcraft

It is redundant in some formulations, but not others. In any case, it's traditional to include it. I'll mention the redundancy on Axiom of the empty set. -- Toby 05:33 Feb 21, 2003 (UTC)

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The singular, "Zermelo-Fraenkel axiom", does not make sense as the title of this article. It makes more sense to title an article "cat" than "cats", and is in accord with Wikipedia conventions, but we're not defining a general concept of a Zermelo-Fraenkel axiom; we're defining a short list of specific axioms and schemas; the whole phrase "Zermelo-Fraenkel axioms" is really a proper noun. It's as if ten separate articles were titled "Comandment" without any article titled "Ten Commandments". The plural in the title of this article makes sense for the same reason the plural in an article titled "Ten Commandments" would make sense. Michael Hardy 22:53 Jan 15, 2003 (UTC)

How about Zermelo-Fraenkel set theory? I'm finding that in metamathematics books more than anything else. -- Toby 05:33 Feb 21, 2003 (UTC)

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Do most mathematicians believe anything about ZF? Most mathematicians use operations on sets, and the validity of those operations in in effect codified by ZF, but I don't think most mathematicians think about ZF, let alone believe anything about ZF. Michael Hardy 23:08 Jan 15, 2003 (UTC)

I agree; most mathematicians couldn't care less about ZF. Sure, the axiom of choice is interesting, but not the axiom schema of replacement or the axiom of well foundation. I've changed it to "metamathematicians", which may not be precisely the right term. -- Toby 05:33 Feb 21, 2003 (UTC)

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"On the other hand, the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal." Does this refer to weakly or strongly inaccessible cardinals? -- Schnee 01:25, 10 Aug 2003 (UTC)

Weakly -- all inaccessible cardinals are weakly inaccessible, while only some of them are strongly so. -- Toby Bartels 22:33, 13 Feb 2004 (UTC)

Just noticed this comment; it's a couple years old, but should probably be addressed for the record. Actually the more common convention is that "inaccessible" means "strongly inaccessible", and if you mean "weakly" you say it explicitly, unless it's clear from context. However from the point of view of Schneelocke's question, it doesn't matter, because the existence of a weakly inaccessible cardinal has the same consistency strength as the existence of a strong inaccessible. --Trovatore 02:40, 29 April 2006 (UTC)

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Shouldn't we state the axioms in their weakest forms, i.e. "if two sets are the same then they are equal" and "there is a set"?

list of set theory topics

Wikipedia has no list of set theory topics! Set-theory mavens, please help. Once it is created (or maybe even before it is created?), it should be added to the list of lists of mathematical topics. Michael Hardy 00:30, 13 Jun 2005 (UTC)

What does the exclamation point mean in logic?

The first order logic articles doesn't list it as one of the symbols, yet it is this article. --212.85.24.83

$\exists !$ means "there exists a unique". --Zundark 15:47, 6 December 2005 (UTC)

Consistency proof (partial, of course)

http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html reads: "Abian (1969) proved consistency and independence of four of the Zermelo-Fraenkel axioms". The original paper is available at http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1093888220. I would really love to add a note about it to this article, but I fear I lack the necessary understanding on the topic.

Is the axiom of infinity consistent with the axiom of foundation?

Basically, we start with the set of all ordinals that do not contain themselves. This set, just like the set in Russell's paradox, is not well defined.The question that comes up is whether it is the set N of all finite ordinals or the set of Burali-Forti paradox.The following reasoning makes me think that it is the set N itself.

For, N={0}U{Sx:x eN},

This is true, but it is not the definition of N.--Aleph4 13:48, 27 January 2006 (UTC)

where S is the successor function ('e' is 'belongs to';! prefixed is the negation).By the axiom of foundation, x !e x for all x.

So, N is {0} U {Sx:x e N and Sx !e Sx}.

But {Sx:x e N and Sx !e Sx} ={y:y!=0,0 e y,y e N,and y !ey}

[The above equality holds only because N is infinite]

So, N={0}U{y:y!=0,0 ey,y e N,and y !ey}.

But the condition y e N is tautological.

No, it is not.--Aleph4 13:48, 27 January 2006 (UTC)

Please see the clarification below.--Apoorv1 06:59, 2 February 2006 (UTC)

So,N={0}U{y:0 e y and y !e y}.

But this set is not well defined.

See also links http://mathforum.org/kb/message.jspa?messageID=3808877&tstart=0 and http://mathforum.org/kb/message.jspa?messageID=4165416&tstart=0). See also[[1]]

--Apoorv1 06:55, 27 January 2006 (UTC)

Those are cryptic comments. A little more elaboration would help.--Apoorv1 05:16, 30 January 2006 (UTC)

In the absence of any further elaboration from your side, these are some additional comments.

'When we set

N={0}U{y:y!=0,0 ey,y e N,and y !ey}, it means

y e N iff y=0 or [y !=0 and 0 e y and y != y and y e N].

Now, for statements p and q, [p<-->p and q]<-->[p<-->q],so

I guess you mean [p <-->(p and q)] in the first bracket. Your formula is not a tautology. The first bracket is equivalent to [ p --> q ]. --Aleph4 10:28, 2 February 2006 (UTC)

The reverse implication is also true in this case, because the axiom of infinity by itself does not guarantee the existence of a set larger then N all of whose member contain 0.. So, in the system 'ZF less the powerset axiom', N is the set of all ordinals containig 0 that do not contain themselves. Alternatively, it is the set of all ordinals containing 0. In any case,in the system 'ZF less the powerset axiom'the set guaranteed to exist by the axiom of infinity will not satisfy the axiom of foundation.If the system 'ZF less the axiom of powerset' is inconsistent, will the system ZF be consistent?--Apoorv1 07:34, 6 February 2006 (UTC)

What do you mean by because the axiom of infinity by itself does not guarantee the existence of a set larger then N all of whose member contain 0.? In particular, what do you mean by "larger"?

1. ZF minus Power set does not guarantee the existence of an uncountable set.
2. ZF minus Power set does guarantee the existence of a set X with the following properties:

• X contains N
• X is not equal to N
• all elements of X (except for 0) contain 0.

(Note that in the context of set theory, N is always understood to contain 0. I hope I have not misunderstood you there.)

Which precisely is this set X that you are referring to ?----Apoorv1 04:59, 18 March 2006 (UTC)

For example, the set omega+1.--Aleph4 08:21, 28 March 2006 (UTC)

X is supposed to be closed under the successor operation. So how is it w+1?In fact, as I have averred earlier, the repeated application of the successor operation can give you no set bigger than N, unles you assume that N !e N, or equivalently S(N) !=N.In the absence of the axiom of powerset and regularity, the membership of N in N cannot be decided.--Apoorv1 08:55, 28 March 2006 (UTC)

There is another way of approaching the issue. As you say,

1. ZF minus Power set does guarantee the existence of a set X with the following properties:

• X contains N
• X is not equal to N
• all elements of X (except for 0) contain 0.

This means, that the axiom of infinity is actually two different axioms:

1A)The set N, containing 0 and closed under successor operation exists.

1B)Another set X, containing 0 and N and closed under successor operation exists.

Once again consider the system ZF minus powerset and regularity and only 1A part of the axiom of infinity.

Then N is the largest ordinal in this system and hence, N eN <--> N!eN .

The addition of axiom 1B or the axiom of regularity or powerset does not help us to resolve this basic contradiction.

--Apoorv1 11:14, 31 March 2006 (UTC)

N is never defined to be the set of all ordinals containing 0 that do not contain themselves. The clause "do not contain themselves" does not make much sense, because whenever x is an ordinal, then x is not an element of x. This follows from the definition of "ordinal".

If the system 'ZF less the axiom of powerset' is inconsistent, then of course ZF is also inconsistent. But you have not shown either of the two statements.

Aleph4 20:45, 16 March 2006 (UTC)

See remarks above.--Apoorv1 06:01, 28 March 2006 (UTC) I moved them to "below" Aleph4 08:21, 28 March 2006 (UTC)

y e N iff y =0 or [y !=0 and 0 e y and y !e y],so

N={0} u {y : 0 e y and y !e y} .

The analogy with

S={y:y e S and y=2} <-->S={y : y=2}

makes the above reasoning clearer'.--Apoorv1 06:59, 2 February 2006 (UTC)

The empty set will satisfy the left equality, but not the right one. --Aleph4 10:28, 2 February 2006 (UTC)

The point is well taken. However, see the comments above.--Apoorv1 07:34, 6 February 2006 (UTC)

Infinite sets

The discussion was getting a bit confusing, so I moved Apporv's question here.Aleph4 08:21, 28 March 2006 (UTC)Just to ensure readability, I have copied the relevant comments of Aleph4 below.--Apoorv1 10:50, 28 March 2006 (UTC)

'ZF minus Power set does not guarantee the existence of an uncountable set. ZF minus Power set does guarantee the existence of a set X with the following properties: X contains N; X is not equal to N;and all elements of X (except for 0) contain 0.' Aleph4

For the moment, let us say that X is some (as yet unidentified) countable set.Now X is infinite only if Sx!=x for all xeX.But Sx!=x iff x!ex.So X ={0 and all x containing 0 that do not contain themselves}. Now consider the system ZF less the powerset and regularity axioms.In the absence of the powerset axiom, P(X) does not exist. S(X) exists only if X !e X. But X is nothing but the set in Russell's paradox and the question whether X e X or X !e X cannot be answered.So X, which is countable by hypothesis,is not well defined. --Apoorv1 06:01, 28 March 2006 (UTC)

It seems that you are claiming various things that you cannot prove. For example, "Now X is infinite only if Sx!=x for all xeX". It seems to me that you are claiming

(A) if X is infinite, then Sx !=x for all x in X.

Or perhaps you meant to say

(B) if Sx != x for all x in X, then X is infinite.

I think we can agree that (B) is false. (e.g., take X empty).

The axiom of regularity implies that Sx != x for all x (because x not in x, for all x). So if you assume regularity, the clause "if X is infinite" is redundant.

Without the axiom of regularity, (A) cannot be shown.

Aleph4 08:21, 28 March 2006 (UTC)

I think we need to see my comments in the context of our discussion.The set X we are talking of is the set guaranteed to exist by the axiom of infinity and is closed under the successor operation. If X is finite, it could not be closed under the successor operation. If X is infinite and Sx=x for some xe X, then x =X and so X e X and SX=X for a countable set, directly in contradiction to the axiom of regularity.

So the only case of interest is the case when Sx!=x (i.e x!ex)for all x in X.

In this case, in the system ZF less powerset and regularity, the set S(X)!=X only if X !e X.Since X = {All x such that x !e x}, X , which is countable, is not well defined.Since X !e X <-->X e X in this system, the addition of the axiom of Regularity to this axiom system cannot remove this basic contradiction. --Apoorv1 10:35, 28 March 2006 (UTC)

Empty Set, Pairing, Subsets are redundant

I draw your attentions to the masterly exposition of Suppes (1972). Suppes dispenses with Empty Set by simply deriving the empty set as the extension of A not equal to A. He then sets out Pairing, and Subsets (Separation) very early on, and delays introducing Replacement as long as possible. But when he does so, he shows that Pairing and Subsets become easy theorems. Thus his definitive listing of the ZFC axioms, on p. nn, does not include Empty Set, Pairing, and Subsets.

It is indeed a revealing fact that the vast majority of working mathematicians don't know any axiomatic set theory and are not curious about it. They are not even interested in the foundations of mathematics. Taking set theory seriously seems limited nowadays to Berkeley, Tarski's students, a number of Israeli and Eastern European mathematicians, and the coterie studying Quinian set theory. The limited interest in set theory and metamathematics nowadays may be largely driven by the lack of interest in those subjects on the part of granting agencies.202.36.179.65 18:00, 27 February 2006 (UTC)

One might say that Paul Cohen killed the foundations of mathematics as classically conceived by answering (in the negative) in 1964 what remained post-1950 as its biggest open problem, whether Choice and GCH followed from the ZF axioms. Thereafter foundations went off in two directions. One direction, the primary outlet for which was then and still is JSL, the Association for Symbolic Logic's Journal of Symbolic Logic, continues to address the progressively more esoteric questions remaining within the ZF framework, of which there are plenty but which the average mathematician finds it harder to relate to as more of them get answered. The other direction is comprised of various new and not so new frameworks whose respective perspectives make the questions raised by classical foundations less well motivated and which instead substitute their own open problems motivated by their own perspective. Proof theory, modern or abstract algebra with an emphasis on universal algebra, and category theory are all active subjects today, each with at most one or two hundred actively contributing participants, each considering itself as addressing foundational concerns in mathematics. But even those are becoming old hat, and today we find a lot of interest in foundational studies of coalgebras (which arose from categorical thinking but which could as well have come from ZF), quantum programming languages (as a much needed sensitization of Birkhoff and von Neumann's old quantum logic to the more quantitative and dynamic aspects of Heisenberg uncertainty and entanglement), concurrency theory (broadly construed to cover any kind of concurrent behavior by fleets of vehicles, packet networks, parallel programs, corporations, orchestras, armies, etc.), and so on. Foundations is far from dead, it just isn't recognizable today if you define it narrowly to be ongoing research into the consequences of the ZF axioms. In view of this diversity, to say that the ZF axioms are the axioms on which mathematics is based today is a bit of a stretch. Sets and functions play a very important supporting role in modern mathematics, but the ZF axiomatization of binary membership, however popular with old-school foundationalists, is by no means the only approach to either foundations in general or sets and functions in particular. The language of ZF is not even mathematically natural: when did you last see anyone make use of a homomorphism that respected set membership? Monotone functions respect order, group homomorphisms respect the group operation, linear transformations respect linear combinations, and gangsters respect membership in the Cosa Nostra, but what morphism has ever respected membership in a set? It is sheer hubris for a relation that can't get no respect to claim to support mathematics. Vaughan Pratt 00:59, 21 August 2006 (UTC)

Right arrow

What does the $\rightarrow$ mean ? Where is it defined ? Thanks. --Hdante 08:44, 5 March 2006 (UTC)

The right-arrow is the symbol for material implication in propositional logic. See Propositional logic#Soundness and completeness of the rules. Otto ter Haar 12:24, 5 March 2006 (UTC)

Symbolism

Where is the cheat sheet to explain what all the symbols mean? Kd4ttc 22:12, 10 March 2006 (UTC)

For those of use familar with mathematics but not expert the symbols in the text are opaque. Any reference that one can go to for the symbol meanings? Kd4ttc 23:02, 12 March 2006 (UTC)

Does first-order logic help? --Trovatore 23:14, 12 March 2006 (UTC)

Yes! I'm thinking of developing a compendium of symbols that the casual reader may browse. Kd4ttc 23:40, 12 March 2006 (UTC)

So there's already Table of mathematical symbols and List of operators (these are probably duplicative as it is). You might take a look at the best way to help people find these. --Trovatore 23:44, 12 March 2006 (UTC)

You are more than kind! Kd4ttc 01:12, 13 March 2006 (UTC)

Syntax and semantics

At the moment the article has syntax and semantics all mixed up. ZFC per se is purely syntax; a collection of strings of characters and rules for manipulating them. Therefore, for example, the first sentence from the introduction,

ZFC consists of a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., all mathematical objects) are sets.

is wrong; ZFC, in and of itself, has no ontology. The set-theoretic notions that (depending on your philosophy) interpret, justify, or motivate ZFC, do have an ontology, but they are distinct from ZFC itself.

(Really it's these set-theoretic notions that should be given the status of "the most common foundation of mathematics", not ZFC itself.) --Trovatore 03:41, 12 March 2006 (UTC)

Are ordinals essential?

I changed the sentence (referring to Zermelo's axioms)

This axiomatic theory did not allow the construction of the ordinal numbers, and hence was inadequate for all of ordinary mathematics.

because it is not true. Most of "ordinary" mathematics can be developed without using ordinals. Transfinite induction does appear sometimes in "ordinary mathematics" (which I understand in this context as "mathematics except set theory", or perhaps "mathematics except mathematical logic"), but usally any well-ordered set of the appropriate length will do, and in fact transfinite induction is usually done along any well-order of the set that is being investigated at the moment (field whose closure is to be computed, Banach space on which a functional is to be extended, etc). Often even transfinite induction is not used, and is replaced by using Zorn's lemma as a black box.

Set theory of course all the time uses von Neumann's cumulative hierarchy, where ordinals and the replacement axiom are quite natural or even necessary.

Aleph4 20:54, 16 March 2006 (UTC)

The silly consistency comment

I refer to the following comment

Because the axioms of Peano arithmetic are ZFC theorems, and the consistency of Peano arithmetic cannot be proved by virtue of Gödel's second incompleteness theorem, the consistency of ZFC cannot be proved using ordinary mathematics.

On its face, this sentence is misleading at best, false at worst. There are many known consistency proofs of PA, Godel's theorem notwithstanding. I plan to fix this soon. CMummert 22:37, 26 April 2006 (UTC)

Yet Another Silly Consistency Comment?

"Because of Gödel's second incompleteness theorem, the consistency of ZFC cannot be proved within ZFC itself. " Surely this should read something more like "Because of Gödel's second incompleteness theorem, the consistency if ZFC could be proved within ZFC itself, then ZFC would be inconsistent"?

Of course, ZFC is consistent, but we don't *know* that (do we?). SinghAgain 17:14, 04 Feb 2007 (UTC)

Distinctness condition missing in formal statement of Axiom of Choice?

Shouldn't there be an assertion that D != B in the statement of the axiom of choice? That is, two _distinct_ elements of A are disjoint. Otherwise, B and D can be the same element of A, in which case the consequent of the innermost implication fails and the axiom as a whole is vacuous. Or have I missed something...? Awmorp 11:33, 28 April 2006 (UTC)

Yes, you're right. CMummert 11:37, 28 April 2006 (UTC)

Limit Ordinals-a matter of faith?

The first limit ordinal is w (omega).It is defined as the union of all preceding ordinals. This presupposes that the set of all preceding ordinals exists-so the definition of w really presupposes the existence of and the definition of w (and is a non definition).

If we consider numbers as primary, then we know that given a number, there is a greater number,but there is no infinite number.Similarly,given a finite set of numbers, we can have a bigger set.But can we have an infinite set?

If we consider the seq. A1={1},A2={1,2},A3={1,2,3} and so on,what is the process by which the limit {1,2,3. . .}is obtained? In particular, there is no metric by which the successive members become closer to each other.So the axiomatic assertion,through the 'axiom of infinity' of the existence of a 'biggest' or 'infinite set w' appears to be a leap of faith,which is quite opposed to our assertion that an infinite number or magnitude does not exist. --Apoorv1 07:47, 12 May 2006 (UTC)

Well, an axiom basically is a formal leap of faith. You can prove that the axiom of infinity cannot be proven or disproven from the rest of ZFC. -Dan 17:11, 24 May 2006 (UTC)

Nitpick about finite axiomatizations

...proved that ZF (and hence a fortiori ZFC) cannot be ... finitely axiomatized.

Why a fortiori? Maybe this is backwards? -Dan 17:11, 24 May 2006 (UTC)

Good point. Anyone want to look up Montague's paper, and see what he actually proved? (We shouldn't reverse them unless he really did prove ZFC not finitely axiomatizable in 1957). --Trovatore 22:27, 24 May 2006 (UTC)

ZFC really is not finitely axiomatizable, because it proves that any finite subset of its axioms has a model. The sentence in the article is misleading. CMummert 11:40, 25 May 2006 (UTC)

I have now fixed that sentence and the false statement about Godel's theorem that I mentioned higher on the talk page. CMummert 12:00, 25 May 2006 (UTC)

Of course I knew that ZFC is not finitely axiomatizable. My question was about what Montague had proved. While it seems most natural that he would have proved the result for ZFC, I haven't actually seen his paper. Have you? --Trovatore 13:53, 25 May 2006 (UTC)

No, I haven't; I left in the reference to Montague only out of respect for the original author, and I would not mind if it were removed. I did look up some papers on Mathscinet before I edited the article this morning. The best bet seems to be MR0163840, which is dated 1961 instead of 1957. Here is a quote from its description of Montague's paper Fraenkel's addition to the axioms of Zermelo with ellipses to indicate where I pruned it.

Fränkel's addition is the replacement (or Ersetzungs-schema (RS). ... The author defines a (countable) subset of the cumulative type structure which can be proved to satisfy all instances of Zermelo's comprehension schema (SSF: schema of set formation); ... And if an additional finite set $A$ of sentences in the notation of set theory is added, a model satisfying both $A$ and SSF can be established. ... A consequence is that RS is not finitely axiomatisable over SSF, and the same holds for any consistent extension of RS ...

That quote indicates to me that Montague did prove the result that the article indicates, although the year may be wrong. Moving to the area of personal opinion, I generally feel that there is little reason to give attribution of results such as this one in wikipedia. Extremely important or extremely difficult results may deserve special attention, but this result does not have those properties. So I would vote in favor of not attributing the result at all in the article, and just pointing out the ZFC is not finitely axiomatizable. CMummert 14:36, 25 May 2006 (UTC)

Switch to Kunen's axioms

I changed the previous set of formal axioms, which I think were correct, to the exact set of axioms in Kunen's book. Here are my reasons:

• Kunen was the first book I could find that gave symbolic forms of the axioms.
• The other set of axioms was unsourced.
• The other set of axiom was typeset poorly. The conventions were not those employed in contemporary literature. Several of the previous axioms were typeset at over 8 inches of width on my monitor. I could not easily tell whether the previous axioms were correct because the typesetting was too hard to read.

I also added references to Kunen and Jech's books. Once I figure out how to do proper inline citations I will fix that. I think that this article should somewhere mention the cumulative hierarchy, which is the fundamental motivation for the axioms of ZFC. CMummert 20:38, 23 June 2006 (UTC)

I have several objections concerning some details in Kunen's list:

1. The axiom of set existence is not really set-theoretical; it is a purely logical axiom (or a consequence of purely logical axioms). I suggest to either omit it altogether (as some axiomatizations do -- e.g. Jech or Fraenkel, Bar-Hillel, Levy), or to replace it by the "axiom of the empty set". The axiom of the empty set follows of course from the axiom of separation, but one needs the axiom of the empty set to deduce the axiom of separation from the replacement axiom.
2. I think that Kunen is unique in calling the "well-ordering theorem" an axiom. He does this only for his own convenience to speed up the development in his book. There are many theorems that are (over ZF) equivalent to the axiom of choice, but only few of them deserve the name axiom (rather than "theorem" or "lemma"). I suggest to use one (or several) of the customary formulations of AC (choice function for P(X), choice function for families of nonempty sets, choice function for disjoint families/disjoint sets of nonempty sets).

--Aleph4 18:39, 24 June 2006 (UTC)

I have made some changes that at least partially address your concerns.

A disadvantage of using Kunen's axioms would be that we are stuck with using them more or less exactly as he phrased them; if we change them, then they aren't Kunen's axioms any more. I did look at Jech's book; he gives the axioms in English, but not in symbolic form. I think it is nice to have a concrete reference for the specific formal axioms that we list, since the axioms are not completely canonical. It seems to fit WP:NOR better. On the other hand, I added English decriptions based on the previous article, and pointed out that those descriptions are not from Kunen.

I have no objection if someone else finds a book that gives formal symbolic statements of the axioms, puts those symbolic statements into the article correctly, and gives a correct citation to them. That is, I am not advocating Kunen over any other source. Kunen's book was just the first book I could find, and I thought it was sufficient.

I think that it would benefit the article more to explain the various ways in which the axioms are not canonical than it would to pick a different noncanonical choice of axioms.

My impression of the distinction between the well ordering principle and the axiom of chice is this. Cantor suggested the well ordering principle (Every set can be well ordered) as an axiom in the 1880s. Some opposition to the supposed obvious nature of this axiom arose, and Zermelo gave a proof which reduced the well ordering principle to the axiom of choice (Every sequence of nonempty sets has a choice function), which Zermelo believed to be conceptually simpler. So the well ordering principle has historical precedent as an axiom.

CMummert 19:14, 24 June 2006 (UTC)

As I understand history (although I do not have a reference at the moment), Cantor did not consider an "axiomatization" of set theory at all. (Does your "1880s" refer to "Über unendliche lineare Punktmannigfaltigkeiten"?)

On the other hand, the main point for Zermelo's 1904 and 1908 papers was to prove the well-ordering theorem as a theorem, and to isolate the axioms used in this proof, in particular the axiom of choice.

Jech's book (millenium edition) gives the axioms in English on the first page, and a few pages later gives formal versions. I like these version better than Kunen's -- not only because the axiom of choice is given in the customary form, but also because he uses lowercase and uppercase variables, which is more intuitive.

Aleph4 19:50, 24 June 2006 (UTC)

Ah. I saw the English ones at the start of Jech's book, looked at Kunen's book, and stopped looking. I have no objection if you would like to switch the article to Jech's axiomatization. CMummert 20:32, 24 June 2006 (UTC)

There are two different types of bi-implication arrows used in the Axioms as set out here. Is there a reason for this or should it be changed? 86.20.228.25 16:12, 4 February 2007 (UTC)

I cleaned that up some; it had drifted since the version originally added. I have no preference at all for Rightarrow vs. rightarrow, so I just chose one. You should feel free to make corrections like this yourself. CMummert · talk 18:03, 4 February 2007 (UTC)

Axiom of extensionality

The English description for this axiom reads "Two sets are the same if and only if they have the same members." But in fact, the axiom as it is presented states that "Two sets are the same if they have the same members" (a conditional rather than a biconditional). I'm not familiar with the original text from which this axiom was taken, but it's presented as an iff in the stand-alone article (axiom of extensionality). Whichever we decide to use on this page, the English description should match the mathematical description. Mathfreq 21:59, 15 August 2006 (UTC)

The fact that sets that are equal must have the same members is a property of equality which is an axiom of the underlying first order logic. Thus only the converse, that sets with the same members are equal, needs to be added as an axiom. More importantly, the formal axioms are directly quoted from Kunen; please don't change them unless you have a reference for the new ones. Anyone is free to change the English text, which is original here, to explain what is going on. CMummert 22:44, 15 August 2006 (UTC)

Axiom of Choice v. Well-ordering Theorem (v. Zorn's lemma)

I'm just wondering why we've labeled the well-ordering theorem with its ZF-logically equivalent "axiom of choice". The well-ordering theorem applies terms such as "minimal" which have no defined context.

Secondly, these axioms are supposed to allow an intuitive basis for ZFC logic. We very well could substitute the axiom of choice/well-ordering theorem with Zorn's lemma. This creates a structurally identical system; however, it would create more confusion.

I have only a little background in logic, so I don't feel comfortable changing this part of the page, but I think it should be reverted to a form more compatible to the name "axiom of choice," if only for the fact that it's more self-contained.

Mo Anabre 20:11, 22 February 2007 (UTC)

The axioms here are taken verbatim from Kunen's book, where he calls the ninth axiom the axiom of choice. I have presented an argument higher on this talk page for keeping the entire set of axioms from one book; see the section Switch to Kunen's axioms.

I still think that it would benefit the article more to explain the various ways in which the axioms are not canonical than it would to pick a different noncanonical choice of axioms. CMummert · talk 21:06, 22 February 2007 (UTC)

I am aware of the discussion and your reasons for replacing it, but it begs several questions, some of which I've already proposed.

(1) If we're to decide on the stability aspect, we probably should decide based on the form most often used. In this way, the well-ordering theorem, while ZF-logically equivalent to the form I've seen used much more often, is not the same thing. This is my point with adding Zorn's lemma as an alternative. If we want to use any ZF-equivalent form and you want it to be as far from the canonical form as possible, let's just change it: list the axiom of choice as "Every non-empty partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element."

(2) The aspect of being self-contained. The axioms shown here should be self-contained (aside from the obvious necessity of defining metalogic/syntax). This specific argument is really what I'm focusing on the most. In my mind, the reason why I refuse to accept the well-ordering theorem as the axiom of choice is that it isn't self-contained. One must access another definition much more advanced than metalogic to find out what "is minimal under R" means.

Personally, if you just wanted references, I'll give you some references. Meanwhile, it seems that making a section on Kunen's axioms (and labelling as such) would be more within the bounds of reason.

Mo Anabre 19:13, 23 February 2007 (UTC)

If you would like to change the article so it uses the ZFC axioms from a different book, then please feel free to fix it; I have no attachment to Kunen's axioms, but I don't see how any other set of axioms is more "canonical" so I have no desire to change the article myself. I explained higher on the talk page why I think all 9 should come from the same book. CMummert · talk 19:48, 23 February 2007 (UTC)

Dash in page title

I moved "Zermelo–Fraenkel set theory" to "Zermelo-Fraenkel set theory" based on Wikipedia:Manual of Style (dashes). I was not aware that there was a discussion of changing the guideline on Wikipedia_talk:Manual_of_Style_(dashes)#Proposed_expanded_version_of_advice_on_hyphens_and_dashes_in_MoS.--Patrick 09:32, 9 June 2007 (UTC)

As a general rule, one should not move an important article like this without first discussing it on the talk page and getting a consensus. I have seen many articles moved the other direction, from hyphens to dashes. So you appear to be going against the trend. JRSpriggs 10:16, 9 June 2007 (UTC)

Weren't those moves in the other direction mostly done by the banned user Jon Awbrey, and weren't they also done without consensus? (That is certainly the case for this article, at least.) --Zundark 11:51, 9 June 2007 (UTC)

There is a general rule that names like this should use endashes. This is one case where JA got it right. The idea is that you use endashes to separate surnames of different persons, but hyphens where it's what Charles Matthews calls a "double-barreled name" of a single person (so Burali-Forti paradox rather than Burali–Forti paradox, because it's named after Cesare Burali-Forti rather than a Burali and a Forti. --Trovatore 23:25, 9 June 2007 (UTC)

Some were moved by Trovatore (talk · contribs) or Blotwell (talk · contribs). JRSpriggs 09:12, 10 June 2007 (UTC)

I don't mind moving it back if that is desired. But currently Wikipedia:Manual of Style (dashes) says that hyphens are preferred.--Patrick 10:46, 10 June 2007 (UTC)

There is an ongoing discussion at that MOS page about several changes to the style guide, including alowing en dashes in titles again. It might be bset to wait for that to resolve. — Carl (CBM · talk) 11:48, 10 June 2007 (UTC)

Now that the policy has been updated, how about we move it back to the correct title with en dash? Dicklyon 05:58, 15 June 2007 (UTC)

Heck, I just did it. RossBot will take care of the double redirects shortly, I bet. Dicklyon 06:01, 15 June 2007 (UTC)

Switch to Shoenfield's axioms: any objections?

Since many of Kunen's axioms are redundant, I propose to use Shoenfield's axioms instead, Shoenfield being a standard book on the subject as well, and probably the most accurate. They consist of: extensionality, regularity, union, power set, infinity, and replacement. (Shoenfield actually uses the "subset" and "replacement" schemas instead of the "union" axiom and "replacement" schema in the main text, but the equivalence of the two sets of axioms is mentioned in the exercises, the latter being more common.) I will change them if there are no objections. —The preceding unsigned comment was added by Neithan Agarwaen (talk • contribs) 16:37, 13 July 2007 (UTC)

Actually, I do kind of object to that. Redundancy, as I see it, is not really a problem. Having aussonderung and replacement listed separately is convenient for explaining the difference between Z and ZF, and I think it's more standard and more historical. It's not a huge deal, but on balance I prefer the status quo. --Trovatore 19:50, 13 July 2007 (UTC)

I'm not sure what "most accurate" means; I know of no serious objections to the overall correctness of Kunen or Jech's books on set theory. The "redundancy" is not a serious issue in my opinion. On one hand, as Trovatore points out, the current axioms are quite historically motivated and well known. Another issue, perhaps less serious, is that it is common to look at structures that don't satisfy replacement or power set, in which case the above list of axioms is no longer obviously enough to prove even pairing. I would have no objection to switching to Jech, although I see no advantage; but I think it would be a disservice to our readers to select an even more idiosyncratic selection of axioms, that will disagree with the majority of undergraduate texts, just to avoid some redundancy. — Carl (CBM · talk) 22:01, 13 July 2007 (UTC)

I agree with Carl. JRSpriggs 01:52, 14 July 2007 (UTC)

Well, I think there are enough objections already. Although I have to say, I do not really understand them. The axioms of Schoenfield are the same than Kunen's, without "set existence", "pairing", and "specification", and these can (and should) be mentioned as being derivable from the other axioms. In fact, I just mentioned Schoenfield so as to provide a standard reference for the axioms ("most accurate" in the treatment of set theory itself, I meant, not as far as the axioms are concerned), as has been asked elsewhere. So no information or clarity would be lost. On the other hand, having a minimal set of axioms not only makes the theory seem more elementary, but it is also greatly useful for the working mathematician, for the same reasons as it is useful to have less primitive symbols and less logical axioms/inference rules in formal systems. Also, Kunen's axiom of choice is quite nonstandard, I think, and the usual axiom "there exists a choice function on any set" is more common. Anyway, I agree that it is not really important. -- Neithan Agarwaen 11:27, 14 July 2007 (UTC)

There are some comments immediately following the last axiom which include identifying the axiom of pairing as redundant. If you want to add to those comments or emphasis them more, that would be helpful, and I hope it would satisfy your concerns. JRSpriggs 03:05, 15 July 2007 (UTC)

Z notation link

Is there any reason to keep the Z notation link and the corresponding category? Z notation is not really related to math. —Preceding unsigned comment added by 75.62.4.229 (talk) 07:33, 22 November 2007 (UTC)

"Union" textual vs. equation

For any set x, there is a set y such that the elements of y are precisely the members of the members of x.

The text says (all) the members of (all) the members. However, the equation says '$\exists D$', that is 'one member'. Is this correct ? I'm tagging the article as contradictory. --Hdante 08:13, 5 March 2006 (UTC)

The axiom of union is here correctly formulated. The axiom defines B := ∪A. B is called the union set of A. B collects precisely all sets C which are member of any set D in A. Otto ter Haar 12:02, 5 March 2006 (UTC)

That's true ! :-) --Hdante 17:26, 5 March 2006 (UTC)

The current formula doesn't say that, though. That formula says that B contains all the sets which are members of sets in A, but it doesn't say it contains only those sets. The formulation given in Axiom_of_union avoids this problem. Rsmoore (talk) 16:22, 17 March 2008 (UTC)

Both formulations are common; there's no problem. The formulation in the axiom of union article follows from the one here and a comprehension axiom. The axioms here are, for the sake of consistency, all taken from a single source (Kunen's book). The text here does not claim that the set A is literally the union, only that it contains all the elements that are in the union. As Kunen points out, this formulation is easier to work with when the goal is to verify that the axiom is satisfied by a particular model. — Carl (CBM · talk) 17:52, 17 March 2008 (UTC)

replacement implies comprehension

The article used to say that the ZFC axioms minus comprehension imply comprehension. Here is one proof that this is true when the empty set is assumed to exist. Let A be any set and assume we want to form the set $W=\{x\in A\mid \phi (x)\}$. Let $Z$ be any set in A such that $\phi (Z)$ (if there is no such set Z then W is the empty set). Let $\psi (x,y)$ say that either $y=x$ and $\phi (x)$ or $y=Z$ and $\lnot \phi (x)$. Thus $\psi$ defines a function from A to A such that any element satisfying $\phi$ maps to itself, and all the other elements map to $Z$. Then the range of $\psi$ is exactly the set that we are trying to construct, and this exists by replacement on the formula $\psi$. This proof doesn't need to be in the article, but I think the statement is interesting. CMummert 23:53, 4 September 2006 (UTC)

The problem is that you are assuming a version of the axiom of replacement which is not the one in the article. The one in the article says:

$\forall A\,\forall w_{1},\ldots ,w_{n}[\forall x\in A\exists !y\phi \rightarrow \exists Y\forall x\in A\exists y\in Y\phi ].$

Notice that there is no limit on how many extra elements one can put into Y and thus it could be that Y might just be the same as A in your example. One needs the axiom of separation to convert this version of replacement into the one which you are assuming. JRSpriggs 08:20, 5 September 2006 (UTC)

While I expect the version stated in the article to be equivalent to the one CM is thinking of, there remains a problem. The description of replacement still states that the generated set Y is the co-domain of the function, which is no longer true. Correct me if I'm wrong, but the cryptic mention of a restriction to avoid paradox can't possibly account for this discrepancy since making the set larger risks creating more paradoxes, if anything.

I rephrased the English description to make it agree with the formal axiom. The cryptic comment about a restriction is referring to the restriction on the free variables of the formula. I don't think the two versions of replacement are equivalent in the absence of comprehension, which is why JRSpriggs's comment settled the matter for me. CMummert · talk 23:52, 27 January 2007 (UTC)

Let me expand on what CMummert said about the restriction on free variables. Y should not appear free in φ. If it did, then the existential quantification over it on the right side of the implication would cause trouble because then the function implicitly defined by φ would change from what it was on the left. JRSpriggs 12:40, 28 January 2007 (UTC)

Huh?

What is this?:

$\forall z[z\in x\Leftrightarrow z\in y]\land \forall z[x\in z\Leftrightarrow y\in z].$

Why is it possible that $z\in x$ and $x\in z$ in the same context? The formula spans over z twice separately, so we cannot protest too much here, but x seems to be the same kind of set, so the first z spans over members of x, which then must be a set, but the second z spans over sets of sets. Is this intended to mean anything, or is it a typo where $\in$ should be replaced by $\subset$? ... said: Rursus (bork²) 13:21, 25 February 2009 (UTC)

Oh, I forgot: in the extension of axiom 1. ... said: Rursus (bork²) 13:26, 25 February 2009 (UTC)

OK, I got it wrong. It is quite possible that x, y and z are sets of sets, which seems to be an assumption in this set theory. ... said: Rursus (bork²) 13:39, 25 February 2009 (UTC)

As the lead says "all individuals in the universe of discourse are such sets". Each thing is a set and all of its elements are also such sets. Thus all the variables and constants in this theory represent sets of sets of sets of sets of ... ad infinitum. If you keep moving down from a set to one of its elements (also such a set), then eventually you reach the empty set which has no element to choose. See pure set. JRSpriggs (talk) 22:26, 25 February 2009 (UTC)

It seems like it would be easy to use two different variables instead of writing z twice.

metamath

I removed this paragraph:

One piece of evidence bearing on ZFC as a foundation of mathematics is Metamath, an ongoing web-based project that seeks to derive much of contemporary mathematics from the ZFC axioms, first order logic, and a host of definitions, with all proofs verified by machine. As of early 2008, the Metamath database includes about 8000 proved theorems. This project can be seen as being in the same spirit as Bertrand Russell's Principia Mathematica, except grounded in logical and nonlogical axioms that benefit from nearly a century of subsequent research.

There's maybe some better way to mention Metamath in the article, but I find the claims in the paragraph above to be a bit overstated. 75.62.6.87 (talk) 20:27, 5 April 2009 (UTC)

σ set theory

I removed a link to the arxiv about an alternative set theory. Because almost anyone can publish almost anything on the arxiv, I don't think we should generally be using preprints there as references. If some published text or journal refers to the theory, that would make me want to consider whether to include it. Are there any references like that? Really I am interested in evidence thqat set theorists other than the author are interested in the concept. — Carl (CBM · talk) 00:42, 16 July 2009 (UTC)

Some problems and some questions

I am finding it difficult to make sense of the WikiPedia specification of ZFC. Below are some of the problems and questions I have. Can some kind reader please help me understand what I have misunderstood. W J Eckerslyke (talk) 12:51, 1 February 2009 (UTC)

Chickens and eggs

ZFC “is the standard form of axiomatic set theory and as such is the most common foundation of mathematics”. Yet its “universe of discourse” is claimed to comprise “all mathematical objects”. How can this be? How can ZFC claim to be based on objects which are supposedly based on ZFC? At the very least some clarification is needed.

What clarification would you propose? — Carl (CBM · talk) 20:02, 1 February 2009 (UTC)

As far as I know, any mathematical object can be modeled in set theory. That is, it can be represented by a set and it properties and relationships can also be represented by sets.

As far as I know, no claim is made that ZFC is based on anything other than logic and sets; and no claim is made that other mathematical objects are necessarily based on ZFC or sets. JRSpriggs (talk) 10:53, 2 February 2009 (UTC)

What about non-mathematical objects?

Why should a set theory be confined to discussion of mathematical objects only? Surely a set theory, even an axiomatic one, could usefully be applied to sets of books, stars, or badgers?

This is true from a certain point of view, but for the mathematical study of set theory, there is no need for sets that contain non-mathematical objects, and indeed no need for set that contain anything other than sets. The effects of including urelements in set theory are well known and generally not of much mathematical interest. — Carl (CBM · talk) 20:04, 1 February 2009 (UTC)

As Carl says, non-mathematical objects could be included by making them urelements. This result in ZFU (Zermelo–Fraenkel set theory with urelements). But ZFU itself can be modeled in ZFC and is not especially interesting to a mathematician; it just introduces fruitless complexity. JRSpriggs (talk) 10:59, 2 February 2009 (UTC)

Individuals

Is it really necessary, or even desirable, to assume that “all individuals ... are sets”? This assumption certainly offers some potential simplification, but it also causes a serious problem which ZFC does not seem to address. The question is whether for all individuals x it is true that {x}=x. If this is not true then all individuals have the same members, i.e. none, and are thus indistinguishable from each other and from the empty set. But if it is true then neither the Axiom of Regularity nor the Axiom of Infinity bears examination!

In set theory with urelements, the axiom of extensionality has to be changed for exactly the reason you describe. This is describe in Jech's book, for example. But since ZFC has no urelements this is not an issue with ZFC. So I don't see how this relates to the present article, which is about ZFC in particular, not set theory in general. — Carl (CBM · talk) 20:10, 1 February 2009 (UTC)

Domain of discourse

Are there any individuals (other than the empty set)? “Many authors require a nonempty domain of discourse as part of the semantics of the first-order logic in which ZFC is formalized.” I cannot see why this should be optional. If there is no given 'domain of discourse' then the whole thing just enables the construction of confections of the empty set.

A very few authors use free logic, but most of them require that a domain of discourse must be nonempty, so they agree with you. But what relevance does this have to the article? — Carl (CBM · talk) 20:08, 1 February 2009 (UTC)

I support "free logic" which is actually merely valid logic. "Logics" which assume the existence of an individual in the domain of discourse are not valid, i.e. not real logic. JRSpriggs (talk) 11:05, 2 February 2009 (UTC)

Active sets?

“The axioms of ZFC govern how sets behave and interact.” Strange wording! Sets are essentially passive objects, and do not behave or interact at all. It seems to me the axioms do no more than specify what sets are deemed to “exist”.

In ordinary English, "behave" and "interact" suggest change and thus the passage of time, which does not apply to ZFC. However, there certainly are relationships between sets which are definable in terms of the element relation. Typically, one thinks of stages in the process of definition as being constructions which occur over time in our minds. So we apply words which, perhaps, are describing our mental state more than the actual mathematical objects. Timelessness is a property of the content of mathematics generally, not just set theory; but mathematicians live in time. JRSpriggs (talk) 11:14, 2 February 2009 (UTC)

Consistency

There seems to be some doubt that ZFC is in fact consistent. Doubtless there is no actual proof of its consistency, but at first glance there really does not seem to be enough in it to allow the possibility of inconsistency. However, its blithe use of reference to “any property” (Axiom 3) and to “any formula” (Axiom 6), without any attempt to limit the scope of the “any”, clearly leaves infinite scope for things to go wrong, and consistency is therefore likely to depend on just what scope is chosen for properties and formulas. It also seems to be left open whether or not “the background logic includes equality” (Axiom 3). The conclusion might perhaps be that ZFC may or may not have any consistent realisations, but it is very likely to have some inconsistent ones.

There is little doubt among mathematicians that ZFC is consistent. In particular, there is no great suspicion that it is actually possible to prove a contradiction from the ZFC axioms. On the other hand (and this point of view dates back to Zermelo), if ZFC is inconsistent, the most likely way to discover that is to rigorously study its conssequences.

Some mathematicians have other objections to ZFC – they might dislike classical logic, or not like infinite structures. But the percentage of such mathematicians in the general population is vanishingly small. — Carl (CBM · talk) 20:07, 1 February 2009 (UTC)

Infinite sets

“The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω.” Indeed, it would seem that ω is the only set whose existence this axiom guarantees, even if there is a non-empty domain of discourse. So perhaps this should be called the Axiom of ω?

The common name for it is the axiom of infinity, and this is the name that all set theory texts I have seen use. The axiom is so named because it directly postulates the existence of an infinite set. In some statements of the axiom it does not literally say that ω exists, only that some set containing ω exists. — Carl (CBM · talk) 19:57, 1 February 2009 (UTC)

Assuming you mean a set containing ω as a subset rather than as an element, you've taken the words out of my mouth. The set could be any limit ordinal, or it could be something like ω union the successor sequence of {{{∅}}}. The obvious way to deduce that ω is a set would be to apply the axiom of specification, except that I can't at the moment see a predicate we could use that is within the ZF axioms. -- Smjg (talk) 12:55, 5 June 2010 (UTC)

Sometimes the axiom of infinity is stated as ``There exists an infinite set" (with careful meaning given to infinite). Then we define ω to be the intersection of all infinite sets. (Note that this intersection exists only if there is at least one infinte set. Otherwise we get nonsense). Setitup (talk) 22:34, 2 March 2010 (UTC)

The intersection of all infinite sets is empty. For example, consider the infinite set consisting of just the even natural numbers, and the infinite set consisting of just the odd natural numbers. One can define ω as the intersection of all infinite sets that contain the empty set and are closed under the map $x\mapsto \{x\}\cup x$. — Carl (CBM · talk) 02:38, 3 March 2010 (UTC)

But the axioms don't enable us to

• know what is meant by an infinite set (that said, that condition is redundant)
• conceive the infinite class of sets having the property and identify the intersection of them all. -- Smjg (talk) 12:55, 5 June 2010 (UTC)

Not clear what you mean by any of this. The intersection of all sets satisfying some predicate P is completely well-specified, given that at least one set satisfies P. And the predicate "is an infinite set" is certainly definable in the language of set theory. Carl is certainly correct that the intersection of all infinite sets is empty.

Your remarks about the axioms "not enabling us" to know or conceive of these things are correct, but maybe not for the reason you think. Axioms don't "allow us to know or conceive of" anything; that's not the function of axioms. Oh, sure, the axiomatic method can derive new knowledge from old knowledge, and such knowledge can be conceptualized. But that wasn't what you seemed to be talking about. --Trovatore (talk) 20:16, 5 June 2010 (UTC)

To Smjg: See axiom of infinity#Extracting the natural numbers from the infinite set for two different ways of extracting ω from a superset of it.

An infinite set is a set which is not a finite set. See the section finite set#Necessary and sufficient conditions for finiteness for a variety of ways of characterizing finiteness. JRSpriggs (talk) 17:03, 6 June 2010 (UTC)

Basically, what I meant is that ZF merely postulates the existence of one set having this property. "The intersection of all sets satisfying some predicate P" is well-specified only if "all sets satisfying some predicate P" is. This, in turn, is well-specified in a set theory that includes the axiom of unrestricted comprehension, but ZF deliberately doesn't include it.

But now I see: a natural number is a set:

• where every element is either 0 (∅) or the successor of a sibling element (x ∪ {x})
• and which itself is either 0 (∅) or the successor of one of its elements (x ∪ {x})

I was trying to do something like that, but on the infinite set as a whole instead of each of its elements. Thanks, that's cleared it up. -- Smjg (talk) 01:14, 10 June 2010 (UTC)

Well, good, but you're still missing a point. You don't need unrestricted comprehension to make the intersection well-specified. The intersection is just the collection of all sets x such that x is in every set y for which P(y) holds. That works just fine to specify the intersection, without no need to collect all such y into a completed totality. --Trovatore (talk) 01:19, 10 June 2010 (UTC)

Oh, maybe you're concerned that there might be no set collecting all such x. But there is, granted that there's at least one y₀ such that P(y₀) holds, because then the desired collection is a subset of y₀. --Trovatore (talk) 01:21, 10 June 2010 (UTC)

So essentially, you're exercising ZF's axiom of specification on the postulated infinite set with the predicate Q(x) "every set containing ∅ and the successor of every one of its elements contains x". Such that you don't need to worry about collecting all y satisfying P(y), only about whether P(y) ⇒ x ∈ y. Have I got that right? -- Smjg (talk) 23:56, 11 June 2010 (UTC)

Power set

There is what seems to be a dangerous imprecision in the presentation of the axiom of power set. The text says “The power set of x is the class whose members are every possible subset of x.” This “possible” may be interpreted as meaning “conceivable”, whatever that means. But that would be wrong. In this context “possible” can only mean “possibly existing in accordance with these axioms”. This is perhaps clearer in the formal definition: $\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y],$ in which the scope of the $\forall$ is more obviously limited to sets allowed under ZFC.

I've reworded this to avoid the word "possible", as this word doesn't seem helpful here, and is potentially confusing. --Zundark (talk) 13:56, 1 February 2009 (UTC)

Thanks for response. However, I had and have no objection to the word "possible", but merely sought clarification of its meaning. Which I still seek. Are you able to confirm that the ZFC power set includes only subsets allowable under the other ZFC axioms? W J Eckerslyke (talk) 17:51, 1 February 2009 (UTC)

I removed the word "possible" because I had an objection to it. As for your question: intuitively, it contains all subsets (formally - see the formula). --Zundark (talk) 19:25, 1 February 2009 (UTC)

The informal goal of the ZFC axioms is to describe statements that are true about the collection of all sets. So the axiom of power set is intended to say that for any set X there is another set containing all subsets of X. I don't think that the statement "only subsets allowable under the other ZFC axioms" actually makes sense if you examine it more closely – because the collection of all sets satisfies each the axioms of ZFC, it's not at all clear that any set is proscribed by the ZFC axioms. — Carl (CBM · talk) 19:59, 1 February 2009 (UTC)

I am not sure it is entirely true that no set is proscribed by the ZFC axioms: the Axiom of Regularity seems to put some (reasonable) limits on what sets may be deemed to exist. But, apart from that, ZFC is indeed short of proscriptions. However, the axioms do prescribe a minimum population of sets which the ZFC adherent is required to believe in, starting with the empty set and the von Neumann ordinal, and including all sets whose existence is guaranteed by virtue of e.g. the Axioms of specification, pairing, union and replacement. You seem to be saying that in addition to this guaranteed minimum population the adherent is allowed to believe in any other conceivable sets which are consistent with the axioms. If that is so then I think it should be made clear in the article. But even then these additional sets must surely be optional, and anything provable under ZFC must surely not be allowed to assume more than the guaranteed minimum population of sets. Nor, I believe, can the Axiom of power set be deemed to guarantee more subsets than are required by the other axioms. Which was my original point. W J Eckerslyke (talk) 10:41, 3 February 2009 (UTC)

It's true that the axiom of regularity restricts the universe of ZFC to well-founded sets instead of arbitrary sets. Kunen discusses this choice at some length in his book.

The minimum model of ZFC is the constructible universe, and thus it is true in some sense that these would be all the sets that one "has to believe in". I'm not sure whether this point really warrants a long discussion in the article, though. In particular, the standard way of describing the power set axiom is that it guarantees that for any set X there is another set Y containing all subsets of X. If you relativize this to a model M you get: for every X there is a Y which contains every subset of X that is in M. But the intended model is the collection of all (well-founded) sets, in which case Y does include all subsets of X. — Carl (CBM · talk) 14:14, 3 February 2009 (UTC)

I added a section on the cumulative hierarchy, which really was a glaring omission. But I don't know exactly what other changes you are proposing. — Carl (CBM · talk) 16:07, 3 February 2009 (UTC)

Let us call a set "founded", if the transitive closure of the singleton of the set is well-founded with respect to the element relation. If a set is founded, then all its elements are founded. If all elements of a set are founded, then the set is founded. If a set is founded, then its powerset is founded. None of the operations for creating sets allowed by the axioms of ZFC can create anything unfounded if applied to things which are founded. JRSpriggs (talk) 03:11, 4 February 2009 (UTC)

I'm just an ignoramus trying to make sense of all this, which I have not yet succeeded in doing, despite the kind assistance of several contributors, for which I am very grateful. I note that "the minimum model of ZFC is the constructible universe", and I assume that this means that "the constructible universe" is the most that anybody proving theorems on the basis of ZFC can rely on. If so, that greatly limits what can be proved on the basis of ZFC, and if true this is a point which would be of sufficient importance to warrant a brief mention in the article, IMHO. The fact that "the intended model" is somewhat larger does not seem to alter this. Moreover, if to enable this larger potential model we allow a set to be anything pure and well-founded that I can imagine then I don't see why we need Axioms #3 to #8 at all: I can perfectly well imagine the existence of such sets without being given explicit licence to do so. W J Eckerslyke (talk) 16:49, 4 February 2009 (UTC)

The constructible universe is the smallest model of ZFC containing all the ordinals. But this does not mean that it is possible to prove in ZFC that every set is constructible, and so one cannot particularly "rely on" constructibility when proving things in ZFC. By analogy, the only elements of a field of characteristic 0 that must exist are the rational numbers (in this context the rationals are the prime field of characteristic 0). However, one cannot assume when proving something about a field of characteristic 0 that every element is rational, and indeed the field of rationals is not a very interesting field of characteristic 0. Similarly, although L satisfies the ZFC axioms, it is not one of the more interesting models of ZFC to set theorists.

Re "I can perfectly well imagine the existence of such sets without being given explicit licence to do so." – keep in mind the axiomatic method. The goal is to isolate a few particular axioms about sets, each of which is intuitively justified, and from which it is possible to prove many interesting theorems. The intuitive justification for the ZFC axioms is the cumulative hierarchy, and the ZFC axioms are strong enough to prove essentially all theorems of non-set-theoretic mathematics. In this sense the axioms of ZFC represent a particularly successful application of the axiomatic method, showing that no further intuitions about the nature of sets, beyond those conveyed by the ZFC axioms, are required for the bulk of mathematics. — Carl (CBM · talk) 18:16, 4 February 2009 (UTC)

Definability

“Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.” Is that a good use of effort? Would it not be more sensible instead to develop a set theory based on a set definition capability, and to accept the existence of all sets which can be defined by it and no others?

It might seem more natural to you, but set theorists have found that the study of the relationshup between definability and the axiom of choice to be very fruitful. In particular, one reason that the constructible universe has many of its properties is that it has definable choice functions for every nonempty set. On the other hand, topological regularity properties such as projective determinacy require that there is no definable choice function on the reals of a certain complexity. — Carl (CBM · talk) 20:01, 1 February 2009 (UTC)

Why no mention of the axiom of the empty set?

nt —Preceding unsigned comment added by 86.146.254.177 (talk) 07:10, 16 December 2007 (UTC)

Presumably Kunen chose to omit it because it can be deduced from the other axioms. From the axiom of infinity we get the existence of a set. From the axiom schema of specification we can reduce that set to the empty set. It is unique by the axiom of extensionality. JRSpriggs (talk) 08:08, 16 December 2007 (UTC)

The current article is incorrect in its treatment of the empty set. In the preamble to the axiom of infinity, The current article asserts the existence of the null set is derived from extensionality, regularity, specification, pairing, union, and replacement. This is false, because each of those axioms are (vacuously) true if no set exists. Hurkyl (talk) 09:19, 1 February 2008 (UTC)

I tried to fix this problem. How do you like the new version? Notice that the axiom of infinity itself supplies the existence of a set which is needed to show that the empty set exists. JRSpriggs (talk) 06:47, 3 February 2008 (UTC)

I don't see anything false, so I'm mostly content. However, I notice another problem. The presentation invokes the axiom of infinity in order to prove the existence of the empty set -- and yet the statement of the axiom of infinity presupposes that the empty set exists! I think it's not possible to simultaneously: write a correct article, keep Kunen's axioms in the form he wrote them, and to omit the axiom of the null set (or whatever equivalent Kunen used). Hurkyl (talk) 03:32, 5 February 2008 (UTC)

First-order logic, with no axioms at all, proves that something exists--if you want to allow the opposite possibility you have to go out of your way to do it. Then by applying separation, you get that the empty set exists. You don't need the axiom of infinity for this. --Trovatore (talk) 03:37, 5 February 2008 (UTC)

That depends on your choice of logical axioms, or semantically speaking, on what you define to be a model. While customary axioms prove $\forall x\varphi \to \exists x\varphi$ or $\exists x(x=x)$, and customary model theory does not consider the empty set as a model, it may be convenient to also consider empty models (e.g, in algebra, when you would like to have a theorem "the intersection of two subsemigroups is a subsemigroup), and hence to use a system of logical axioms in which "there exists something" is not derivable.

In set theory we are of course not interested in empty models; all the more reason to explicitly postulate the empty set, just to make this point clear.

Since the ZFC axioms are anyway not independent of each other, I think that the derivability of the empty set axiom is not a good reason to exclude it. (The fact that Kunen excludes it, on the other hand, is a good reason. Just not a very good one.)

A reason to include it is the role that it plays in the relationship between replacement and separation. Replacement in its strict form (namely, that the class of function values is a set, not only contained in a set), implies separation in the case that the set $\{x\in z:\varphi (x)\}$ is nonempty, but you need the empty set axiom to get the full separation axiom from replacement. (Of course all this is not relevant when we use the ZFC axioms, but it might be worth a remark when we discuss them.)

Aleph4 (talk) 12:13, 5 February 2008 (UTC)

See also Talk:Axiom of empty set#Does logic imply the existence of the empty set?. JRSpriggs (talk) 09:24, 6 February 2008 (UTC)

Of course, I should have checked there... Aleph4 (talk) 14:00, 6 February 2008 (UTC)

Trovatore is right in that I didn't consider shunting the postulate that something exists off to the logical axioms -- but as Aleph4 points out, that postulate only exists in certain formulations of first-order logic. If the intent is for the article to rely on this postulate, then I assert that it needs to be explicitly stated someplace.

The problem, as I see it, is that the choices made in the article has many drawbacks. The restriction to a certain kind of logic is unnecessary, it complicates the exposition, and it diverges from traditional presentations of ZFC. The benefits are... fewer axioms?

The previous discussion sounds like Kunen's list of axioms actually did include an axiom asserting the existence of a set. Am I reading that correctly? Hurkyl (talk) 02:51, 14 February 2008 (UTC)

Well, what is standardly called "first-order logic" does imply that something exists. Modifications of FOL that avoid that implication are called free logic. --Trovatore (talk) 03:10, 14 February 2008 (UTC)

Really, the problem is with how the axiom of infinity is presented here, with the condition $\varnothing \in X$. But we can rewrite it without referring to the empty set, namely by stating instead that there exists some element of X of which nothing is an element, i.e. $\exists x[x\in X\land \forall y(y\notin x)]$. I'll see what I can do.... -- Smjg (talk) 11:43, 9 August 2010 (UTC)

(←) We should not rewrite the axiom. The axioms are taken directly from a source for verifiability. Of course we could come up with dozens of equivalent axiom systems, but then they are not the systems that we can actually verify by looking at a source. I have, in the past, checked the axioms symbol by symbol against the source.

Apart from that, on a technical level, your change wouldn't address any issues with the existence of the empty set. The symbol $\emptyset$ is just an abbreviation, it is not actually in the language of ZFC. The formula you changed to is just an unwrapping of the abbreviation. It doesn't actually change the formal axiom, and so it cannot address any perceived problems. In other words, Kunen's axiom is an abbreviation for the axiom you changed to.

More generally, I don't see what issue you are concerned about with the empty set. The issue of "at least one set exists" is covered in detail in the second paragraph of that section. — Carl (CBM · talk) 12:15, 9 August 2010 (UTC)

Perhaps a better way of translating $\varnothing \in X\,$ into the language of set theory is $\forall x(\forall y(y\notin x)\rightarrow x\in X)\,.$ Then if the empty set exists, the axiom says that X is a superset of the natural numbers; and if the empty set does not exist, the axiom says that it does exist, to wit $\varnothing =X\,$ in that case. JRSpriggs (talk) 06:50, 10 August 2010 (UTC)

That's pretty. But either translation gives that the axiom of infinity stated here implies that the empty set exists. The article points our that (since the axiom starts with an existential quantifier) the axiom implies some set exists regardless of what the matrix says, and we can use comprehension to make the empty set out of that. Then, empty set in hand, we can go back to the axiom of infinity to get an infinite set, at which point it doesn't matter too much how we translate "$\emptyset \in X$". — Carl (CBM · talk) 11:04, 10 August 2010 (UTC)

The second paragraph states that the axiom of infinity is how we find that any set exists in the first place. As has been said, all other axioms are vacuously true in a system where no sets exist at all.

The beginning of that section states "The following particular axiom set is from Kunen (1980)." It later states that Kunen included the axiom of the empty set, but goes on not to list it. As such, the axiom of infinity as written uses ∅ before we have any concept of what it means. If we're going to discard something from the source, we can't just blindly leave the rest of the source intact without regard for whether it still makes sense.

The point is that the way it is written at the moment is circular logic: existence of ∅ → axiom of infinity → existence of a set → existence of ∅. The formulation I gave doesn't rely on the preconceived notion of the empty set - it merely postulates that X has some member having no members. Therefore, it avoids the circularity. -- Smjg (talk) 13:01, 10 August 2010 (UTC)

Kunen does not include any axiom of the empty set – you must not have actually looked at the reference. He simply assumes at least one set exists, along with the axioms here. Usually, the assumption that at least one object exists is considered part of the background logic. Kunen explicitly says on p. 10:

"Under most developments of formal logic, this is derivable from the logical axioms and thus redundant to state here, but we do so for emphasis."

Now, the axiom of infinity stated here already implies "at least one set exists", so anything that can be proven from the axioms here + "at least one set exists" can also be proven from just the axioms listed here. But really you don't need any non-logical axiom to assert that at least one set exists, it's just part of first-order logic.

In any case, simply changing the matrix of the axiom of infinity cannot make any difference. The key point, from the point of view of "at least one set exists" is that the outermost quantifier is existential. There is no circularity because the axiom of infinity is assumed, not proved. — Carl (CBM · talk) 13:48, 10 August 2010 (UTC)

Sorry, I misread the beginning of paragraph 2. So Kunen implicitly assumed at least one set exists, and then derived the existence of the empty set from this and the axiom of specification. But assuming a non-empty domain of discourse is by no means universal. Then it states that the axiom of infinity is how we know that any set exists (and can therefore derive the existence of the empty set), and this is the only way in the absence of it being assumed that we know any set exists. As such, it's anomalous that we have the axiom of infinity written in a way that assumes we already know ∅ exists.

The axiom of infinity being assumed makes no difference. It can't rely on the existence of something that isn't known to exist yet. Maybe I should've been more specific: existence of ∅ → that the axiom of infinity makes sense → axiom of infinity → existence of a set → existence of ∅.

Or are you taking the view that the definition of ∅ is really embedded in the axiom, like this?

$\exists X\left[\{x\in X|x\neq x\}\in X\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

-- Smjg (talk) 15:09, 10 August 2010 (UTC)

(1) Yes, the definition of $\emptyset$ is already embedded in the axiom, because the symbol $\emptyset$ itself is not actually part of the language. Similarly, the language of ZFC does not contain set-builder notation. The only non-logical symbol in the language of ZFC is the ∈ symbol. There are no constant symbols in the language and no term-forming operations.

(2) The second paragraph does not say quite what you are describing. It says that at least one set exists by background assumptions, and also the axiom of infinity asserts that a set exists. The word "also" is key: these are separate reasons that it isn't necessary to add another axiom asserting that at least one set exists.

The assumption that at least one object exists is nearly universal in first-order logic, to the point that systems that do not have this property have a special name, free logic. Kunen himself says "most developments", rather than "some developments". Also, note my comment below about Jech's book. — Carl (CBM · talk) 16:22, 10 August 2010 (UTC)

I've just reworded the phrase about the axiom for the existence of a set, because I found it unclear (I also reworded the part about the axiom of choice because it was even more unclear, and could easily be repaired, but that is a separate matter). I did this in complete ignorance of this discussion, so maybe I've unknowingly stepped on some sensitive toes; please don't take offence but just correct. Just as a matter of personal opinion: I think it is a serious problem that the axiom of infinity should contain a symbol for the empty set, which is not part of the language, because the whole point of giving formal expressions is to remove any doubt about formalization. The question remains (given that set builder notation cannot be used) is the infinity axiom can be written properly in formal notation. Could one just replace $\varnothing \in X$ by $\exists Y\left[Y\in X\land \forall Z(Z\in Y\Rightarrow \lnot Z=Z)\right].$? But I guess this is similar to something already said above, Marc van Leeuwen (talk) 10:45, 27 April 2011 (UTC)

It's worth noting that the other natural candidate for an axiom system to use here, Jech's Set Theory, does not include either the axiom of the empty set or the axiom "at least one set exists" but Jech does point out that the latter is a consequence of the axiom of infinity. — Carl (CBM · talk) 13:52, 10 August 2010 (UTC)

I agree that as it currently stands (2 Jan 2012) the axioms are erroneous because they rely on the axiom of infinity to guarantee the existence of at least one set, but that axiom itself relies on the existence of the empty set. One way to fix that is to include an axiom that asserts the existence of the empty set, as Wolfram does and Stanford (http://plato.stanford.edu/entries/set-theory/ZF.html). The current version is neither fish nor fowl, as it appears to wish to stick as closely as possible to Kunen's version, yet it concedes that Kunen does include an axiom of the empty set. In any case, there's no point in simply importing definitions wholesale from a text without a critical overlay because every text contains some errors (the writers of such texts all being human like us).

A concise solution to the circularity would be to include the assertion of the existence of at least one set in the axiom of infinity, as follows:

$\exists X\left[\exists z(z\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

This can be readily shown to be equivalent to the existing version together with an axiom asserting the existence of the empty set.

But regardless of what solution is adopted, it cannot be left as it is, as the current presentation of the axiom of infinity uses the empty set symbol without definition, so the axiom is a malformed formula. Andrewjameskirk (talk) 20:26, 1 January 2013 (UTC)

The existence of at least one object (therefore at least one set, in this case, because all objects of discourse are sets) is a logical validity in first-order logic (though not in free logic) and follows from no axioms at all. --Trovatore (talk) 20:30, 1 January 2013 (UTC)

To Andrewjameskirk: As explained above, the subformula using the empty set symbol can be regarded as a shorthand for a more correct formula. As such it is more readable than the more formal expression.

Please show how you would derive the usual form of the axiom of infinity from your variation of it! JRSpriggs (talk) 08:28, 2 January 2013 (UTC)

To JRSpriggs: That would be fine if the shorthand is explained. At present it is not explained, which makes the formula, as presented, incorrect, because the symbol $\varnothing$ is undefined. I propose the following simple amendment that allows retention of the shorthand but explains it, in order to make it unimpeachably correct:

[add the following words after the formula]

This axiom appears to presuppose the existence of the empty set $\varnothing$, but it need not do so. In a formulation that does not include an assertion of the existence of the empty set (or of any set other than the infinite set), the subformula $\varnothing \in X$ can instead be considered as a shorthand for the more precise subformula $\forall y(\forall u(u\notin y)\Rightarrow y\in X)\,.$.

Just to see what it would look like, I'll put here what the axiom would look like with the substitution in it, fully expanded.

$\exists X\left[\forall y(\forall u(u\notin y)\Rightarrow y\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

I agree it's a bit long. So the best solution seems to be to keep the abbreviated formula and explain the shorthand, per my proposal here.

Although a slightly more concise version is available by prenexing the $\forall y$ to get:

$\exists X\forall y\left[(\forall u(u\notin y)\Rightarrow y\in X)\land (y\in X\Rightarrow S(y)\in X)\right]$ Andrewjameskirk (talk) 00:05, 22 August 2014 (UTC)

To Trovatore: Some but not all presentations of FOP logic include an assertion of a non-empty domain of discourse. It is unnecessarily restrictive to assert in the article that it is always the case in first-order logic that the domain is non-empty. A harmonious and inclusive way to deal with this would be to state at the top that the axioms are presented under the assumption that the underlying logic has a non-empty domain, and that where that is not assumed, adding an empty set axiom (as per SEP or Ito) can achieve the same result. Regardless of what approach is taken there though, the passage commencing "Second, however, even if ZFC is formulated in so-called free logic, in which it is not a theorem that something exists, the axiom of infinity (below) asserts that an infinite set exists" is circular, as it justifies the existence of the empty set via the Infinity Axiom, which itself relies on the existence of the Empty Set.Andrewjameskirk (talk) 02:43, 8 January 2013 (UTC)

If the empty set is a model, then your logic is not what is standardly called first-order logic. First-order logic has no empty models. Free logic is a different thing; it is not first-order logic as the term is standardly understood. --Trovatore (talk) 02:52, 8 January 2013 (UTC)

The axiom of infinity is not circular; the symbol $\emptyset$ is not actually in the language of ZFC, and so the axiom of infinity does not literally use this symbol. The expression we write down is simply an abbreviation for the actual axiom, which says something like

$(\exists x)[(\exists y)[y\in x\land (\forall z)[z\not \in y]]\land (\forall w)[w\in x\to (\exists v)[v\in x\land v=w\cup \{w\}]]$.

Even that last formula is still an abbreviation; the full axiom would spell out "$v=w\cup \{w\}$" in the language of ZFC. The key point, though, is that the axiom starts with an existential quantifier, so regardless of its internals it implies that at least one set exists. — Carl (CBM · talk) 12:44, 8 January 2013 (UTC)

Zermelo-Fraenkel-Skolem set theory

To my knowledge, the more inclusive name has it's third contributor - Skolem - included. The article should be changed accordingly, in my view. See: [2]. --151.202.103.127 (talk) 21:20, 12 January 2008 (UTC)

See WP:COMMONNAMES. --Trovatore (talk) 21:21, 12 January 2008 (UTC)

Equality without equality

The defining properties of equality are reflexivity (x=x) and the substitution property. The substitution property says that if x=y, then any predicate containing x implies the result of replacing some (or all) of the occurrences of x in the predicate by y. From these one can also show that equality is symmetric and transitive.

If the logic being used does not provide equality, then it can be defined as a macro (abbreviation) for the substitution property. In the case where there is only the one binary relation, ∈, this becomes ∀z[z∈x↔z∈y] ∧ ∀z[x∈z↔y∈z] . This definition works regardless of the axioms used.

However, Palnot has been putting in ∀z[z∈x↔z∈y] which is not the full definition. True, it is equivalent in the presence of the axiom of extensionality (when suitably formulated). But we should not make a definition dependent on an axiom being present. JRSpriggs (talk) 11:35, 15 April 2008 (UTC)

Rephrasing "The Axioms" intro, and Axiom of Infinity

I think it's again worth bringing up the article's inconsistent handling of the empty set and infinity. I cite three specific problems:

• The article claims to present Kunen's axioms, but to the best of my knowledge, Kunen does include an axiom 0 asserting the existence of a set. Previous revisions of the page stated that that axiom was omitted, but that seems to have been removed.

• The statement of the axiom of infinity assume makes use of the empty set symbol, which has not been previously defined.

• The discussion following infinity asserts that infinity (together with other axioms) imply the empty set, making the presentation circular. Also, given the current convention that 'first-order logic' implies the existence of an object (which is Kunen's omitted axiom), infinity is not needed to prove the existence of the empty set. (which is the original reason why the axiom was stricken from the Wikipedia article)

I think the best solution would be to actually include Kunen's axiom 0, and follow it with comments that the axiom is redundant given Wikipedia's conventions, and define the empty set there. But due to the fierce resistance against it seen in the past, I offer the following compromise:

Use this as the introductory paragraph: (addition in bold)

There are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel et al. (1973). The following particular axiom set is from Kunen (1980). Kunen's axiom 0 asserting the existence of a set is implied by Wikipedia's conventions regarding first order logic, and omitted from the list below. The axioms per se are expressed in the symbolism of first order logic. The associated English prose is only intended to aid the intuition.

and use this as the discussion following infinity (after mentioning omega)

... $\varnothing$ is defined to be the unique set satisfying $\forall x:x\notin \varnothing$. It's existence is assumed by this statement of infinity, and can be proven from the axiom of specification and Wikipedia's convention that first-order logic asserts a set exists. Its uniqueness follows from the axiom of extensionality.

--Hurkyl (talk) 02:17, 29 April 2008 (UTC)

This is partly an argument over what First-order logic is, and partly an argument over how to present the axiom of infinity clearly to the reader.

I disagree with having an existential assumption built into first order logic. Thus I believe that some axiom must explicitly affirm the existence of a set (in order for this to be the ZFC we know and love, which does have sets). The axiom of infinity does assert the existence of a set (intended to be an infinite set) — a set which contains any empty set which might exist and any successor of any set which might be a member of this infinite set.

The article states the axiom as:

$\exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

Many people would be able to (sort of) understand this version of the axiom who would be unable to understand a more correct version such as:

$\exists x\left[\forall e(\forall k\lnot (k\in e)\Rightarrow e\in x)\land \forall y\forall z([y\in x\land \forall w(w\in z\iff (w\in y\lor w=y))]\Rightarrow z\in x)\right].$

This more correct version does not assume the existence of the empty set nor of the successor of a set, but once one has it one can prove that they do exist. JRSpriggs (talk) 14:20, 29 April 2008 (UTC)

There's the third option,

$\exists x\left[\exists e(\forall k\lnot (k\in e)\land e\in x)\land \forall y\forall z([y\in x\land \forall w(w\in z\iff (w\in y\lor w=y))]\Rightarrow z\in x)\right].$

I think we can finesse the issue of free first order logic by again pointing out that Kunen (and ZFC, really) assumes that there is at least one set, and that in some formalizations this is built into the semantics of FOL.

Really, I am not fond of emphasizing the analysis of which axioms imply which other axioms. While this is of technical interest, it's only a minor point in the study of ZFC as an axiomatic framework for set theory, because ZFC does include all the axioms, regardless whether they are independent or not. So I favor solutions that don't make the reader dwell on independence too long. — Carl (CBM · talk) 15:22, 29 April 2008 (UTC)

I tend to agree with both of you. In my humble opinion, the absolute most important things are correctness and reader friendliness; my intent was a compromise with what I perceived as a resistance against reintroducing redundancy in hopes of getting an improvement accepted by the community. But if I'm just imagining things, I'd be happy to draft up a bolder suggestion. --Hurkyl (talk) 03:21, 1 May 2008 (UTC)

I added a paragraph to discuss the existence of at least one set - I am relatively satisfied by that. I also moved and lengthened the discussion of the existence of the empty set. I am less happy with that, I hope others can improve it. — Carl (CBM · talk) 12:14, 1 May 2008 (UTC)

Actually, you can prove it from Infinity and Separation without resorting to assumptions based on your interpretation of FOL (this is occasionally useful if we want to base a view of set theory on intuitionistic logic rather than bootstrapping it off of something that already uses an implicit definition of sets). Infinity is the only axiom that begins with an existential, so it cannot be satisfied in an empty universe of discourse. This means at least one set exists. Apply a contradictory formula to Separation, and voila: the empty set exists. TricksterWolf (talk) 07:49, 26 November 2011 (UTC)

Meta theory

The meta theory of anything seem to require us to have a preconception of natural numbers, set union, and subset, and also proof by induction and some other stuff. Also we are allowed to quantify over formulas. So it seems to me that the meta theory of any logic is actually the natural language version (when we read the symbols out loud) of ZF.

That's more or less right. Metatheory is standardly done in ZF (or some sub- or super-theory thereof), although usually in a rather informal style, as you note. You can, in particular, do the metatheory of ZF in ZF (or perhaps a stronger theory like ZF + "There is an inaccessible cardinal" if you want to be able to construct a model of ZF). When this is done carefully and in great formal detail, formulas, and sets thereof, are coded as sets and such metatheoretic relations as satisfaction are then defined as relations on these sets. See, for example, the text _Set Theory: An Introduction to Large Cardinals_ by Frank Drake. TXLogic

From the few things I do understand it is no way as powerful as ZF, e.g. we don't use the AC in any meta theorems, but it does seem a bit circular as ZF is supposed to make rigorous the conceptions above. I don't think it's possible to overcome this circularity without taking things like union and induction, especially natural numbers, to be true primitive knowledge like food and water. Is this how mathematicians think? Money is tight (talk) 15:44, 14 January 2010 (UTC)

There is no circularity. When formulas are coded as sets, formal systems and their models are just complex set theoretic objects. TXLogic

What use is this article?

I've never seen an article on Wikipedia that was so useless in explaining the idea to the layman is simple language. If teh editors of this article are as clever as they would wish to be seen they could have made this MUCH clearer. —Preceding unsigned comment added by 217.206.81.100 (talk) 13:39, 5 March 2010 (UTC)

Concrete suggestions are very welcome. — Carl (CBM · talk) 13:46, 5 March 2010 (UTC)

Perhaps that for which he is asking is an explanation for each axiom of why it is needed to do the usual things that mathematicians do. For example:

• Extentionality defines what a set is, the simplest structure which can be used to categorize things.
• Foundation ensures that a set is what it is intended to be by making sure that its transitive closure can only be implemented in one way.
• Empty set gives us something with which to start to build the Von Neumann universe. So that we have something rather than nothing.
• Pairing allows us to to make ordered pairs and then define functions as sets. It also allows sets to be combined before taking unions.
• Union allows sets which have been put together with pairing or replacement to be unified so that all the elements can be accessed together.
• Separation allows us to pick out the elements in which we are interested and make them a distinct set so they can be further manipulated.
• Power set allows us to combine all the subsets made with separation (or otherwise) into one object, which we could not otherwise do. It gives us access to higher cardinalities which we could not otherwise reach.
• Replacement allows classes of the same cardinality as an existing set to be made a set. It gives us the range of a function which might not otherwise be possible. It establishes the existence of larger ordinals than could otherwise be obtained. It enables definitions by transfinite recursion.
• Infinity gets us out of V_ω and into the interesting stuff.
• Choice assures us that the universe of sets has a relatively predictable structure consistent with some of our intuitions derived from finite sets.

How is that? Is that what you want? JRSpriggs (talk) 03:40, 6 March 2010 (UTC)

Well-ordering Theorem vs. Axiom of Choice

The Wellordering Theorem (WO) and AC are of course equivalent in ZFC. But I think that AC should be in the drivers seat (i.e. in the "List of Axioms" instead of WO). I have no profound reasons for this, but nonetheless, here they are:

1.) The name "The Axiom of Choice" fits better together with the other Axioms. (Important? No, but it is potentially confusing for the beginner to use WO.)

2.) Arguably, AC is much more "intuitively true" than WO - at least in my mind.

3.) The first part of the article speaks about AC and not WO.

An argument for WO instead of AC is that the reference to Kunen's book (he uses WO I suppose) needs modification.

An other argument (for or against) would be historical precedence.

YohanN7 (talk) 09:00, 8 July 2010 (UTC)

I agree — Kunen's choice is a bit idiosyncratic, I think. It does have some practical advantages in that it's often easier to check directly whether a given model satisfies WO, and a historical argument could be made on the grounds that Cantor proposed as a "law of thought" (he didn't call it an axiom) that every set can be wellordered. But then Cantor was not directly responsible for ZF.

Maybe we could take Jech as the source for the axioms instead. --Trovatore (talk) 09:11, 8 July 2010 (UTC)

B t w, a technicality: Shouldn't minimal be least in the definition of a wellorder?

Clearly, if Cantor proposed WO, that is a heavyweight argument for WO. The practical advantages of WO that you mention, on the other hand, are surely known by those actually working with that kind of problems?

Do you agree that AC is easier stated than WO?

I don't suggest a complete rewrite (contentwise) of the subsection in question, merely to swap "drivers". YohanN7 (talk) 09:44, 8 July 2010 (UTC)

On the history of things: Here is a quote from Axiom of choice: "In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle.". YohanN7 (talk) 10:20, 8 July 2010 (UTC)

The list of axioms here is taken from Kunen's book. There are benefits to taking all the axioms from one text, one of which is verifiability of the formal axioms. We could certainly switch wholesale to the axioms by Jech or some other set theory text. But we would want to go through and change all the axioms to fit that book, not just change one of them. However, Kunen's book is the standard textbook in set theory these days, so it's not an unreasonable choice. It just seems like a matter of taste which form of the axiom of choice it taken as an "axiom". — Carl (CBM · talk) 10:57, 8 July 2010 (UTC)

I'm quite happy as things are in the article for the present (was never really unhappy about it). Trovatore and Carl have good arguments (I do not have a copy of Jech's book I should say (Jech's axioms <==> Kunen's axioms)). A change might perhaps (or not) improve the article but the effort might not be worth it.

[That technical detail by the way, I still want "least element", not only "minimal";)]

It is, as Carl says, probably a matter of taste whether one prefers WO over AO. Still, this page is high priority isn't it? It should be since ZFC is pretty much math as it is taught. It should be as good and easy as possible. My personal biased opinion is that AC is a clearer statement for people having little background in math. I think that the statement of AC (biased or not) is clearer to the newcomer than WO that requires relations, linear orders on top of that, etc. /Best Regards YohanN7 (talk) 12:50, 8 July 2010 (UTC)

I just want to second the argument about naming. Having a theorem among a list of axioms is in complete contradiction with what one is doing. At first reading I was actually convinced that the list had ended, and that the section about WO was discussing the status of this theorem relative to the axioms, and it required going back and re-reading a lot of context to get me convinced that the Well-ordering theorem was actually part of the lists of axioms. (Another point that put me off is that the formal expression for the Well-ordering theorem has a rather different appearance from the others, and is, um... informal.) Couldn't it just be called "Axiom of well-ordering" without committing an infraction against Kunen? Marc van Leeuwen (talk) 11:52, 27 April 2011 (UTC)

To Marc van Leeuwen: Axioms are theorems; and it is often a matter of taste which theorems are chosen to be axioms. JRSpriggs (talk) 12:47, 27 April 2011 (UTC)

Axioms are theorems only in the strict sense of the theorem/non-theorem distinction in formal logic. But certainly this is not the reason that the well-ordering theorem is called a theorem. And looking at the theorem article, a theorem is something that requires a proof, and even if one cares to restate an axiom as a theorem with a trivial proof quoting the axiom (don't know what that is good for, but certainly possible), that does not wipe out the distinction. I agree that in an axiomatic approach one can exchange certain axioms for other ones, which implies changing axiom/theorem labels, and this is what goes on here. If one chooses to make well-ordering an axiom, then one should call it that. Marc van Leeuwen (talk) 10:00, 4 May 2011 (UTC)

The word "theorem" is part of the name of the well-ordering theorem. If we talked about the "well-ordering axiom", no one would know what we were talking about (or they would confuse it with the well-ordering principle which is different as it applies only to subsets of the natural numbers). JRSpriggs (talk) 14:31, 4 May 2011 (UTC)

Well, Kunen calls it the axiom of choice, not the wellordering theorem. If we're really following Kunen, we should call it the axiom of choice, even if we give the statement of the wellordering theorem. I agree with Marc that it's confusing to list axioms and call one of them a theorem. (It would be nice to call it something like wellordering axiom but we're not allowed to make up names.) --Trovatore (talk) 17:48, 4 May 2011 (UTC)

I do not think that we should follow Kunen (or any other author) so slavishly that we use the wrong name for a proposition. JRSpriggs (talk) 18:25, 4 May 2011 (UTC)

Well, nevertheless it is confusing to have theorem in the name. Honestly I do kind of think this is an argument for going with Jech's enumeration instead; as I said I think Kunen's is a bit idiosyncratic. There are decent pedagogical reasons for Kunen's choice (it's an easier proposition to check in the context of a given model, and the equivalence can be proved in the exercises) but in an encyclopedia article I kind of think we should use the form more usually thought of as one of the axioms. --Trovatore (talk) 20:24, 4 May 2011 (UTC)

I don't oppose switching to Jech's list of axioms. I do feel there is a benefit to having exactly the list of axioms from some reference, rather than our own list. Unfortunately, there's no "canonical" list of axioms; different authors pick different specific axioms, and even the names of the axioms are changed from one author to another (e.g. separation v. comprehension, replacement v. collection). So the best we can do is pick some author and use that list. — Carl (CBM · talk) 12:19, 6 May 2011 (UTC)

Kunen calls it "Axiom 9. Choice." The statement here is just like the one from Kunen, including the words "well orders" in the formula. The axiom was retitled from "axiom of choice" to "well ordering theorem" in our article in this edit.

In YohanN7's comment, he or she mentions "math as it is taught". That was the reason we took the axioms from Kunen's book in the first place, because it seems to be the standard textbook in set theory these days. — Carl (CBM · talk) 12:49, 27 April 2011 (UTC)

It looks messy the way it is, talking about Axiom of Choice and then... where is the Axiom of Choice? I'd suggest at least something like "9. Well-ordering theorem (equivalent to Axiom of Choice)". --Hugo Spinelli (talk) 21:00, 3 May 2011 (UTC)

For what it's worth my two pence is that I think it should be switched to the Axiom of Choice. Most introductory texts I've seen refer to the Axion of Choice. On this note, it seems peverse that the article is titled Zermelo-Franklen Set Theory then refers throughout to ZFC (the C standing for choice!), perhaps Zermelo-Franklen Set Theory (with Choice) or something. In terms of the earlier reference to keeping it all in reference to one text, I would suggest that we do so but switch to a text that uses Choice not WO, and might suggest Goldrei, "Classic set theory : a guided independent study". Whilst this change would appear to be a big undertaking all it would mean is changing WO to axiom of choice and adding the empty set axiom and keeping the other 8 the same, so it would also solve the discussion higher up this page about excluding the empty set axiom. — Preceding unsigned comment added by 129.67.169.195 (talk) 20:37, 2 June 2012 (UTC)

The problem is, we want to have a sourceable collection of axioms. Right now, the article uses the ones from Kunen. I have in the past looked commented favorably on the possibility of changing to the ones from Jech, but we can't really just go in and substitute ones that we can see personally are equivalent in context. --Trovatore (talk) 21:42, 2 June 2012 (UTC)

TXlogic here: I myself think the case for changing axiom 9 to an explicit statement of Choice is rather overwhelming. Kunen's text is excellent but his decision to axiomatize ZFC with the W.O. Theorem (and, moreover, to call it "Choice") is out of the ordinary, to say the least — of the dozen+ set theory texts I own, Kunen's is the only one that doesn't adopt a standard formulation of Choice. As for the importance of verifiability, Jech's text is certainly every bit as authoritative as Kunen's and, with the exception of Choice, his statement of the axioms is basically identical to Kunen's. I volunteer to make the change; I just don't want to go to the trouble only to have it reverted, so I'm checking here first. Jech uses essentially the formulation "Every set of nonempty sets has a choice function", which is probably the easiest version to state, so I would set the axiom up by defining the notion of a function (and hence that of an ordered pair). On a related note, I would also suggest changing all of the ugly gifs generated by the use of the math tags with proper HTML symbols. I also volunteer to do this. (TXlogic · talk) —Preceding undated comment added 20:09, 25 November 2012 (UTC)

As long as all the axioms are changed to the ones from Jech I definitely won't revert it. My concern is just that we should have a verifiable set of axioms from one text rather than cobbling them together from different places. I don't really think the case is very overwhelming, though, which I why I never made the change myself. — Carl (CBM · talk) 20:17, 25 November 2012 (UTC)

With some minor reservations I support this change. My view is that, while Kunen's choice to make the wellordering theorem an axiom and call it Choice is defensible for his own book, it is not very standard. My reservations would have to do with some other details that I don't quite remember (for example, does Jech talk about the axiom of collection rather than the axiom of replacement? That's not very standard either.) --Trovatore (talk) 22:05, 25 November 2012 (UTC)

Jech uses the term "Replacement". "Collection" (usually) signifies any of several similar but different axioms. For example, the version in KPU does not require a functional mapping on a given set A, but rather only a total mapping φ(x,y), in the sense that, for every element a ∈ A, there has to be at least one object b such that φ(a,b). Collection then says that there is a set B containing at least one such object b for every a ∈ A. Note that B could contain all sorts of other stuff as well, another big difference with Replacement. This version of Collection obviously implies Replacement. (KPU also requires that φ be ∆₀, but that's neither here nor there.) (TXlogic · talk)

a few rough spots

1) The axiom schema of collection appears exactly once in the text, when it is defined. The axiom schema of replacement is discussed extensively, but not listed as an axiom. The relation between the two is not specified within the text of the article; you have to click "axiom of replacement" (or construct a proof) in order to get the story.

2) The axiom of pairing, as listed in the article, does not state that there is a set which contains *exactly* two elements, contrary to the text. The version of the axiom of pairing which is linked to, however, does.

3) (edit: Removed as it was an error on my part)

4) In general, I believe that axioms which are redundant in ZF but not in Z belong in their own labelled section. Since most such axioms are consequences of the axiom schema of replacement, which is perhaps the most unintuitive of the ZF axioms, a few short but explicit derivations would IMO improve the article for set theory outsiders. I think it would be acceptable to alter the numbering of Kunen's axioms as this does not significantly impair verifiability.

5) There should be an explanation of the use of "a set that contains" rather than "a set that is equal to". Presumably this distinction is important when the axiom schema of specification is not assumed, but then we're not in ZF, and as it stands I think it just adds confusion for many readers.

I'm happy to try my hand at contributing, but given that the text and links seem to be at odds with the (contentious but supported, according to the talk page) axioms as written, and the fact that I am a set theory outsider, it seems best to solicit guidance first.

99.37.200.117 (talk) 03:24, 23 August 2010 (UTC)

The collection/relacement thing is mildly irritating - Kunen calls it the "replacement scheme", but some people are overly picky about "replacement" versus "collection". I edited the text there to keep the word "collection" but also include the word "replacement". I don't think it's really helpful to focus on which axioms are redundant to which others, even to the extent that we currently do. — Carl (CBM · talk) 17:37, 29 August 2010 (UTC)

Errors, Inaccuracies, and Infelicities in this Entry

1. The article is entitled "Zermelo-Fraenkel set theory" but the entry begins by talking about ZFC. "ZF" should be introduced to indicate the axioms without Choice. "ZFC" should be introduced as ZF + Choice.

2. "ZFC has a single primitive ontological notion, that of a hereditary well-founded set." ZFC is a mathematical theory, i.e., a set of axioms in a first-order language (at least, in the article — there are of course higher-order versions as well). Its ontological commitments are the stuff of philosophy and belong in an entry on the philosophy of mathematics, not here. The entire notion of an ontological primitive is out of place. The only appropriate notion here is that of a primitive of the *language* — as in fact noted in the following paragraph. It seems to me that, at the least, the first sentence of this paragraph ought to go.

3. "Most of the ZFC axioms state the existence of particular sets." False. Only the axiom of infinity does. The rest of the axioms (with the exception of extensionality) tell you that certain sets exist GIVEN certain others.

4. "In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. This axiomatic theory did not allow the construction of the ordinal numbers." Completely false. In fact, Zermelo did not construct what we now think of as the ordinals, which were defined by Mirimanoff and later, and to greater effect, by von Neumann. But the ordinals are perfectly able to be defined in in Z set theory: An ordinal is a transitive set that is well-ordered by membership. What IS true is that, in Z set theory, one can't prove that there are any ordinals ≥ ω+ω. But that is another matter altogether.

5. "Many authors require a nonempty domain of discourse as part of the semantics of the first-order logic in which ZFC is formalized." Misleading at best. It is a *requirement* of the semantics of first-order logic that the domain of discourse be nonempty. Logics that allow non-empty domains are so-called FREE logics. No set theory text in existence uses anything but classical first-order logic. (Bencivenga published a paper on set theory with free logic, but it's little more than a curiosity.)

6. "Also the axiom of infinity (below) also implies that at least one set exists..." True but irrelevant to the point. That something exists (hence, in the context of ZF, that a set exists) is simply a theorem of the underlying first-order logic of ZF set theory.

7. Set abstraction {x ∈ y | ... } is introduced without explanation or any formal underpinnings in the definition of the empty set. It would not take much to clarify and justify the notation on the basis of Separation and Extensionality.

Also both {...:...} and {...|...} are used. Consistency would be good. 86.132.216.26 (talk) 05:43, 13 August 2017 (UTC)

8. Upper case variables suddenly appear on the scene in the Union axiom. Granted, Kunen does this himself, but the effect is confusing to the reader.

9. As others have noted, it is just nutty to introduce an axiom called the Well-ordering THEOREM. The axioms of a first-order theory are those sentences of the language of the theory that are simply given without proof. The theorems of a theory are those statements of the theory that are provable from the axioms. Granted, every axiom is trivially provable from itself, but it is highly confusing (and, to my knowledge, completely unprecedented) to use the word "theorem" as part of the name of an axiom. It is very confusing. Kunen calls it "Choice". Why you are so dedicated to preserving every detail of Kunen's axiomatization and yet depart from him in this entirely confusing and nonstandard way is baffling. But, even more to the point...

10. As others have also noted, it is simply a really bad idea to use the well-ordering theorem as an axiom. The main reason is that it utterly destroys the self-contained character of the entry. The proper background for the making sense of the well-ordering theorem requires the entire set theoretic analysis of binary relations, including the notion of an ordered pair, which makes no appearance at all in the article. The slavish devotion to the precise details of Kunen's axiomatization basically undermines the usefulness of the article, and it could be completely avoided by using a standard version of Choice, e.g.,

 ∀x[∀y(y ∈ x → (y ≠ ∅ ∧ ∀z((z ∈ x ∧ z ≠ y) → y ∩ z = ∅))) → ∃z∀y(y ∈ x → ∃!w(w ∈ y ∧ w ∈ z))]

where ∃! is the quantifier "there is exactly one".

There is discussion of a version of Choice that says there is a choice function for every nonempty set of nonempty sets, but this has the same liabilities as using the well-ordering version of Choice, namely, that it requires the notion of a function to be defined or the presentation again fails to be self-contained.

11. "At stage 0 there are no sets yet." More accurately, stage 0 is the empty set, at least on most standard presentations of the cumulative hierarchy.

12. "It is provable that a set is in V if and only if the set is pure and well-founded." No notion of "purity" has been defined, nor is one necessary. All sets are "pure" in ZF, i.e., free of urelements in their transitive closures. More seriously, the statement in question is NOT provable, at least, not in ZF — as noted at the beginning of the article, ZF does not countenance proper classes. Of course, reference to V could be explained away as shorthand for talk of formulas, but that once again will require a lot of background that is missing from the entry (and would arguably not be appropriate in this sort of entry anyway).

13. "It is possible to change the definition of V..." But V was never defined in the first place; how is it possible to change a definition that was never given? Furthermore, it is just wrong to characterize the constructible universe as a change in the definition of V. In fact, it is critical to the axiom of constructibility that the definition is NOT changed; otherwise "V = L" would be trivial.

14. "NBG and ZFC are equivalent set theories in the sense that any theorem not mentioning classes and provable in one theory can be proved in the other." The correct way to say this is that NBG is a *conservative extension* of ZFC, not that the two theories are equivalent.

15. The point of Abian and LaMacchia's work is unclear — is its significance the fact that it demostrates the independence of certain of the axioms relative to others?

16. "Because non-well-founded set theory is a model of ZFC..." This is incoherent; well-founded set theory is a THEORY, not a MODEL. What is intended here, I think is that the *consistency* of non-well-founded set theory (relative to ZF) shows that the axiom of regularity is independent of the other ZFC axioms.

17. "If consistent, ZFC cannot prove the existence of the inaccessible cardinals that category theory requires." Inaccessible cardinals are introduced here without definition. And the allusion to category theory is completely out of place; it means nothing to the average reader. And why bring category theory into play at all, as if that were the only motivation there is for introducing inaccessibles? Inaccessibles came on the scene in the history of ZFC LONG before category theory was a gleam in MacLane's eye (and they are profoundly important in Zermelo's 1930 analysis of the cumulative hierarchy).

Finally, virtually all browsers in the world support HTML symbols. The ugly graphical formulas should be removed in favor of HTML. — Preceding unsigned comment added by TXlogic (talk • contribs) 22:19, 19 February 2012 (UTC)

To TXlogic: Most of these complaints are just nit-picking. Some are significant, but need to be dealt with individually. Notice that you are free to try to fix these problems yourself, but others may revert you or further change your work.

Sure, whatever. I'm not sure what counts as a nit and what doesn't, it's a sliding scale. I simply listed errors, inaccuracies and other problems as I found them. I will try to find the time to fix the more significant problems. (TXLogic)

More importantly, your version of the axiom of choice is mistaken. It would be satisfied by any model of ZF, just take z to be the union of x. You left out the essential feature that not more than one element is chosen from each element of x. For correct statements of the axiom of choice, see Talk:Axiom of choice/Archive 5#Unclear. JRSpriggs (talk) 05:58, 20 February 2012 (UTC)

You are of course correct — at one point there was a uniqueness quantifier ∃! there but it seems I managed to omit it in the course of editing my comments. Hm, but come to think of it, that alone wouldn't have been quite right, as you need to ensure that the members of the original set are disjoint. I've fixed the statement above accordingly; I also removed the assumption that x is nonempty, which is not necessary. But, now that I look at it, it might be better after all to use the version that asserts the existence of a choice function, which is much simpler (and, FWIW, is the version I always use, which I suppose I mention as a weak sauce excuse for borking the version I had originally written out above):

 ∀x(∀y(y ∈ x → y ≠ ∅) → ∃f(Function(f) ∧ ∀y(y ∈ x → f(y) ∈ y)))

Of course, "Function(f)" and the notation "f(y)" would have to be defined. (TXLogic)

I agree with User:TXlogic that talk about "ontological notions" is a bit misplaced. Perhaps "primitive notion" or "undefined notion"? Tkuvho (talk) 09:02, 20 February 2012 (UTC)

I think it is best to avoid the suggestion that ZFC, the mathematical theory, contains any "notions" at all; ZFC is simply a certain type of mathematical object, a formal system. Of course, you want to say something about what the intended meaning of the sentences in that system. But I think all you want to say is that the language of set theory contains a single primitive 2-place predicate designed to express the relation "is a member of". (TXLogic)

The purpose of the second paragraph is to clarify that ZFC does not deal directly with urelements nor proper classes, unlike some other set theories. At the same time, this article should ideally (if it was perfect) cover all aspects of ZFC, including philosophical ones, not only mathematical aspects. I think that someone unfamiliar with ZFC would likely find the second paragraph informative in terms of explaining the motivation behind the theory. — Carl (CBM · talk) 13:49, 20 February 2012 (UTC)

"primitive ontological notion" could be replaced by "primitive (undefined) notion" which is clearer. Talking about ontology tends to imply a commitment to "ontological reality" of set-theoretic concepts which is unnecessary and sometimes harmful. Tkuvho (talk) 13:54, 20 February 2012 (UTC)

It is true that there is only one undefined notion, that of a "set", but the paragraph is saying that that undefined notion is intended to represent, not just that the undefined notion is named "set". The intention (of those who developed ZFC) is that ZFC is supposed to capture the cumulative hierarchy of hereditary well-founded sets, when this is viewed as a pre-existing structure outside of ZFC. Kunen discusses this some on pp. 8-9. (Similarly, Euclid's axioms are intended to formalize, among other things, the pre-existing collection of lines on the Euclidean plane, although "line" is an undefined term in the theory itself.) — Carl (CBM · talk) 14:03, 20 February 2012 (UTC)

The fact that "set" is intended to "represent" is already implied by "primitive". I agree with what you wrote but "ontological" is still not as good as "undefined". Tkuvho (talk) 14:06, 20 February 2012 (UTC)

Perhaps "unanalyzed" is better than "underfined"? Tkuvho (talk) 14:07, 20 February 2012 (UTC)

I am OK with "primitive" instead of "ontological". The trouble is the the notion of set was both defined and analysed before the ZFC axioms were created, and the axioms were intended to capture this notion. So when we say "undefined" it could sound like we mean "undefined in the metatheory" when we really mean "undefined in the object theory". The same thing happens for numbers: the concept of a natural number was both defined and analyzed before the Peano axioms were created, but these axioms treat "number" as an undefined term. In modern language we don't often say "undefined term" in the latter sense, we just specify the signature of the formal language. — Carl (CBM · talk) 14:12, 20 February 2012 (UTC)

It is not true that SET is an undefined notion in ZFC (if we must talk of notions); there simply is no such notion in ZFC. The only undefined notion is MEMBERSHIP. The undefined, i.e., primitive, notions of a theory are those intuitively expressed by the primitive, non-logical vocabulary of the theory, the axioms of which are designed to capture those notions formally. SET is a primitive notion in ZFCU, i.e., ZFC with urelements, because there is actually a non-trivial predicate "SET" which is axiomatized accordingly. But there is no such predicate, and there are no such axioms, in ZFC. So, while I think it would be natural and helpful to point out that, INTUITIVELY, sets are the things taken to be in the range of the quantifiers of ZFC, I think it would be confusing to say that SET is an undefined notion of the theory and thereby disconnect the idea of the primitive NOTIONS of a theory from the primitive non-logical VOCABULARY of the theory. (TXLogic) —Preceding undated comment added 19:10, 20 February 2012 (UTC).

Eliminating the notion of "set" from this page will not improve readability. What you seem to be suggesting is that this page should focus uniquely on syntax and avoid discussion of sematics altogether. Formally speaking this may be preferable, but not practically speaking. In principle we could replace every occurrence of "set" by "widget" and speak of ZFC as the theory of widgets. Such an approach would be mathematically rigorous and untainted by ontological considerations, but would it be any more useful than defining a function as a "correspondence" rather than a "rule" in the lead of function (mathematics)? Tkuvho (talk) 08:52, 21 February 2012 (UTC)

Nortexoid's four edits on 24 October 2012

Nortexoid (talk · contribs) made four edits to this article on 24 October 2012 (UTC). I reverted them. He challenged this on my talk page. My problems with his edits are as follows:

• Generally, details which might be useful to some readers were removed in several cases.
• The axiom of infinity (perhaps in combination with the axiom of separation) does imply the existence of the empty set. Yet the statement to that effect was removed. However one expands the axiom of infinity to account for the constant symbol for the empty set (by adding "$\land \forall w(\lnot w\in \varnothing )\,$" at the end, by replacing "$\varnothing \in X\,$" with "$\exists z(\forall w(\lnot w\in z)\land z\in X)\,$", or by replacing it with "$\forall z(\forall w(\lnot w\in z)\rightarrow z\in X)\,$"), it will still imply the existence of that set.
• Stressing that the uniqueness of the emptyset means its independence of "w" may help some readers who would otherwise assume that there is merely one emptyset per "w".
• He said "... the axiom schema of separation implies the existence of at least one set from which the empty set can be 'separated'." which is false. The axiom schema of separation alone does not imply the existence of anything.
• How would one say that a set has an infinite number of members? The method used in the axiom of infinity is at least as good as any other method. So using "immediate consequence" is pointless and confusing.

JRSpriggs (talk) 07:09, 26 October 2012 (UTC)

• Not if there are urelements. The axioms then need to be relativized to sets. I see now, however, that there is some strange passage in the intro which states "The axioms of ZFC prevent its models from containing urelements" which is obviously false without any specific formulation of ZFC having yet been given. And of course there are formulations of ZFC with urelements. The statement should perhaps say instead that the formulation of ZFC given in the article does not allow for urelements.
• Not charitable, since it is was fairly clear that I meant in conjunction with the other axioms. The ellipses in the passage you quote was filled with "The axiom of the empty set is implied by the nine axioms presented here since..."
• This point is hardly important, but in any case the axiom asserts the existence of *one* particular infinite set, viz. omega. Since that set is infinite, the axiom implies the existence of *at least* one infinite set. But it does not "more colloquially" assert the existence of at least one infinite set. There are better candidates for that statement such as "there is a set X, proper subset Y of X, and one-one correspondence between X and Y".

— Preceding unsigned comment added by Nortexoid (talk • contribs) 19:31, 26 October 2012‎ (UTC)

To Nortexoid: Please do not mix your comments in with mine. It is very hard to read and confusing. I have separated them out to avoid the appearance that I have gone mad and argued with myself. JRSpriggs (talk) 04:31, 27 October 2012 (UTC)

• ZFC-with-urelements is not ZFC. It is a separate theory with different axioms. Thus the statement that ZFC forbids urelements remains true.
• No, it was not clear what you 'meant'. In any case, your statement is still wrong because it says that the axiom of separation is used twice when it is only used once.
• You have not defined infinite set, but rather Dedekind-infinite set (a simpler concept).

JRSpriggs (talk) 04:47, 27 October 2012 (UTC)

• The ZF developed in Supppes permits urelements. See esp. p. 20. To say that it is thereby not ZF is a strange remark.
• It says the axiom is used 'twice'? As a native English speaker, I don't get that reading. I sort of see what you mean, but as I said, that is simply not a charitable reading of the sentence, nor one anyone would likely hear.
• Since we are talking about ZFC, emphasis on 'C', the two are equivalent. Take a look at the article you link to which states that in the presence of choice, a set is infinite iff it is Dedekind-infinite. I should add that I said "a better candidate" in any case, so even if you didn't like Choice, it was just an example of a sentence which, for some notion of infinity, literally says there is an infinite set. The axiom of infinity still doesn't say that, I'm afraid.

Nortexoid (talk) 08:51, 27 October 2012 (UTC)

Whatever Suppes does, if a system of set theory admits urelements then it is not the theory "ZFC" described in this article - "ZFC with urelements" is a very different theory.

I looked at the diff for the four edits just now, and I do not see that it was an improvement overall to the previous text. Parts of it seemed less clear, e.g. "The existence of the empty set is unique", but most of it just seemed to be a rephrasing.

In general, apart from these edits, I don't think it's worth spending too much time on how to prove "some set exists". It is a side note only that certain axioms imply this, since the underlying logic already implies it, and because nobody studies empty models of set theory. — Carl (CBM · talk) 18:12, 27 October 2012 (UTC)

• ZFC with urelements just is not a "very different theory" since it's just ZFC with the axioms relativized to sets and, usually, and separate axiom for the empty set. Also, and as I've already stated, the article should just say that it's not interested in urelements instead of stating something false. You're right though--it's not a big deal. But still. Nortexoid (talk) 22:47, 27 October 2012 (UTC)

The second paragraph of the lede begins, "ZFC is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of ZFC refer only to sets, not to urelements (elements of sets which are not themselves sets) or classes (collections of mathematical objects defined by a property shared by their members).". I can't think of a much more direct way for the article to say that. — Carl (CBM · talk) 23:25, 27 October 2012 (UTC)

That paragraph makes it sound like no formulation of ZFC could have urelements. It needs to be phrased more cautiously. Nortexoid (talk) 10:31, 28 October 2012 (UTC)

In fact, Zermelo's 1908 (very informal) axiomatization included urelements. That said, it seems to me that the article should reflect modern usage. The fact is that "ZFC" is used pretty much universally among set theorists and logicians to refer to axiomatizations that have only the membership predicate as primitive and, hence, whose quantifiers are intended to range over pure sets only. On that understanding, it is indeed the case that no formulation of ZFC could have urelements. When urelements are allowed (i) a new predicate (for either sets or urelements) is introduced, (ii) an axiom is introduced saying that only sets have members (and in some formulations, an axiom saying that urelements exist (indeed perhaps "proper class" many, as in Harvey Friedman's formulation), (iii) most of the usual axioms of ZFC are qualified so they only apply to sets, and (iv) the new theory is referred to as "ZFCU" or "ZFC with urelements". As JRSpriggs emphasizes, when you alter ZFC to accommodate urelements, you no longer have ZFC, you have a different (albeit importantly related) theory. TXLogic —Preceding undated comment added 16:50, 19 November 2012 (UTC)

Without the axiom of separation

TXlogic (talk · contribs) says that we need neither the axiom of the empty set nor the axiom schema of separation. He thinks that the axiom of infinity will give us the empty set. But that is not so clear to me. Please show how one can prove the existence of the empty set, if one:

• omits the axiom of the empty set;
• omits the axiom schema of separation; and
• uses the axiom of infinity in the form $\exists X\left[\forall z(\forall w(\lnot w\in z)\rightarrow z\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

It appears to me that to get the empty set from this version of infinity without using separation, one would need to establish the existence of a transitive set (such as the transitive closure of the infinite set X) and then apply the axiom of regularity. Can you show that without first using separation to extract the true natural numbers from X? JRSpriggs (talk) 04:57, 23 November 2012 (UTC)

My guess is they were thinking of the axiom of infinity as

$\exists X\left[\exists z(\forall w(\lnot w\in z)\land z\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

I looked for it but I don't see the claim you mention in the article. — Carl (CBM · talk) 12:38, 23 November 2012 (UTC)

TXlogic (talk · contribs) here. JRSpriggs (talk · contribs): As Carl notes, I never claimed that the existence of the empty set follows from the axiom of infinity. I only claimed that the existence of some set follows from it (trivially, in first-order logic (since it's a logical theorem) and immediately in free logic (since "a set exists" follows immediately from "an infinite set exists"). However, just to respond, of course you need the axiom schema of Separation (or Replacement) to prove that the empty set exists from the existence of an arbitrary set (infinite or not). The empty set axiom is superfluous in the context of the other axioms (as clarified in the exposition of Separation in the article); it would be redundant to add it as an axiom of ZFC because it is derivable from Separation (or, in free logic, Separation+Infinity). TXlogic

I am sorry for bothering you with my mistake. I think that I saw your sentence "Hence, there is no need for a separate axiom." (referring to the axiom of the empty set) and misread it as "... there is no need for an axiom of separation.". I was reading the diff and the variation in colors made it hard to concentrate on the content. JRSpriggs (talk) 05:57, 24 November 2012 (UTC)

Now that you mention it, that sentence could use some sharpening up. TXlogic —Preceding undated comment added 19:15, 25 November 2012 (UTC)

Power set axiom

Chetrasho (talk · contribs) inserted mistaken information on the power set axiom. When I reverted him, he reverted me, saying "Superset is an undefined term linking to subset. A power set is a SET and not a class. Also, copied corrected axiom from axiom of power set".

• However as the text before the list of axioms says "The following particular axiom set is from Kunen (1980).". So we are not free to use just any version of the power set axiom (such as the one at axiom of power set). We must use the particular version in Kunen which is as the article stated before Chetrasho's change.
• "Superset" is a well-known term for the correlative (inverse) of "subset", that is, A is a superset of B just when B is a subset of A. It is defined clearly in the article on subsets.
• To get the version of the power set axiom in the article on it, you must apply the axiom schema of separation to Kunen's version to remove any excess elements which are not subsets of the base set. Until you do that, you only know that the power "set" is a subclass of the set whose existence is guaranteed by Kunen's version of the power set axiom.

Therefore, I will revert again. JRSpriggs (talk) 02:13, 28 January 2013 (UTC)

Edits

Thanks for explaining about Kunen. Sorry for missing out on that part. I still thought the section was unclear, so I made the following edits based on your input:

• Subset is clearly defined and used throughout.

• Kunen's Axiom of Power Set is clearly stated.

• The power set is clearly defined using the axiom of power set and the axiom schema of specification.

Chetrasho (talk) 16:00, 5 February 2013 (UTC)

— Preceding unsigned comment added by Chetrasho (talk • contribs) 15:56, 5 February 2013 (UTC) 

Confusing edit by Arthur Rubin

In [3] Arthur Rubin claimed I added the text "in which you cannot prove that something exists" when in fact according to Wikiblame it was added by User:Crasshopper in this edit. Additionally he appears to have accidentally reverted CBM's edit with no explanation. (Update: he reverted the revert, looks like edit conflict - I made a minor fix to complete CBM's edit). Dcoetzee 14:49, 3 September 2013 (UTC)

My apologies. I still don't think your edits are particularly good, but I'm an expert, so I may not understand the nuances as seen by non-experts. — Arthur Rubin (talk) 15:37, 3 September 2013 (UTC)

The edits are intended to make the material more accessible for people with only a basic entry-level introduction to set theory (and who tend to be confused by first-order logic statements with many quantifiers). I invite any specific improvements or criticism you can raise. Dcoetzee 15:41, 3 September 2013 (UTC)

"It is not a theorem that X" is jargon. Crasshopper (talk) 04:04, 4 September 2013 (UTC)

Missing reference

Shoenfield (1977) is cited twice with no explanation of what it is. 86.166.163.239 (talk) 13:15, 19 May 2015 (UTC)

Presumably, that's

J. R. Shoenfield, Axioms of set theory,

in

Barwise, K. J. Editor (1977) Handbook of Mathematical Logic. xi+1165 pages ISBN 0-7204-2285-X.

But we really should ask user CBM (talk · contribs) who added the section on the cumulative hierarchy in 2009, I think (diff). – Tobias Bergemann (talk) 10:15, 28 August 2015 (UTC)

Yes, that's right. No idea where the reference might have gone, or why I wouldn't have put it in. Good spot, IP. — Carl (CBM · talk) 10:44, 28 August 2015 (UTC)