Ordinal collapsing function

In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals.

The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordinal is brought "from above" to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e., defining a system of notations up to the Bachmann–Howard ordinal).

The use and definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinal analysis, since the large countable ordinals defined and denoted by a given collapse are used to describe the ordinal-theoretic strength of certain formal systems, typically[1][2] subsystems of analysis (such as those seen in the light of reverse mathematics), extensions of Kripke–Platek set theory, Bishop-style systems of constructive mathematics or Martin-Löf-style systems of intuitionistic type theory.

Ordinal collapsing functions are typically denoted using some variation of either the Greek letter (psi) or (theta).

An example leading up to the Bachmann–Howard ordinal

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The choice of the ordinal collapsing function given as example below imitates greatly the system introduced by Buchholz[3] but is limited to collapsing one cardinal for clarity of exposition. More on the relation between this example and Buchholz's system will be said below.

Definition

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Let   stand for the first uncountable ordinal  , or, in fact, any ordinal which is an  -number and guaranteed to be greater than all the countable ordinals which will be constructed (for example, the Church–Kleene ordinal is adequate for our purposes; but we will work with   because it allows the convenient use of the word countable in the definitions).

We define a function   (which will be non-decreasing and continuous), taking an arbitrary ordinal   to a countable ordinal  , recursively on  , as follows:

Assume   has been defined for all  , and we wish to define  .
Let   be the set of ordinals generated starting from  ,  ,   and   by recursively applying the following functions: ordinal addition, multiplication and exponentiation and the function  , i.e., the restriction of   to ordinals  . (Formally, we define   and inductively   for all natural numbers   and we let   be the union of the   for all  .)
Then   is defined as the smallest ordinal not belonging to  .

In a more concise (although more obscure) way:

  is the smallest ordinal which cannot be expressed from  ,  ,   and   using sums, products, exponentials, and the   function itself (to previously constructed ordinals less than  ).

Here is an attempt to explain the motivation for the definition of   in intuitive terms: since the usual operations of addition, multiplication and exponentiation are not sufficient to designate ordinals very far, we attempt to systematically create new names for ordinals by taking the first one which does not have a name yet, and whenever we run out of names, rather than invent them in an ad hoc fashion or using diagonal schemes, we seek them in the ordinals far beyond the ones we are constructing (beyond  , that is); so we give names to uncountable ordinals and, since in the end the list of names is necessarily countable,   will "collapse" them to countable ordinals.

Computation of values of ψ

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To clarify how the function   is able to produce notations for certain ordinals, we now compute its first values.

Predicative start

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First consider  . It contains ordinals   and so on. It also contains such ordinals as  . The first ordinal which it does not contain is   (which is the limit of  ,  ,   and so on — less than   by assumption). The upper bound of the ordinals it contains is   (the limit of  ,  ,   and so on), but that is not so important. This shows that  .

Similarly,   contains the ordinals which can be formed from  ,  ,  ,   and this time also  , using addition, multiplication and exponentiation. This contains all the ordinals up to   but not the latter, so  . In this manner, we prove that   inductively on  : the proof works, however, only as long as  . We therefore have:

  for all  , where   is the smallest fixed point of  .

(Here, the   functions are the Veblen functions defined starting with  .)

Now   but   is no larger, since   cannot be constructed using finite applications of   and thus never belongs to a   set for  , and the function   remains "stuck" at   for some time:

  for all  .

First impredicative values

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Again,  . However, when we come to computing  , something has changed: since   was ("artificially") added to all the  , we are permitted to take the value   in the process. So   contains all ordinals which can be built from  ,  ,  ,  , the   function up to   and this time also   itself, using addition, multiplication and exponentiation. The smallest ordinal not in   is   (the smallest  -number after  ).

We say that the definition   and the next values of the function   such as   are impredicative because they use ordinals (here,  ) greater than the ones which are being defined (here,  ).

Values of ψ up to the Feferman–Schütte ordinal

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The fact that   equals   remains true for all  . (Note, in particular, that  : but since now the ordinal   has been constructed there is nothing to prevent from going beyond this). However, at   (the first fixed point of   beyond  ), the construction stops again, because   cannot be constructed from smaller ordinals and   by finitely applying the   function. So we have  .

The same reasoning shows that   for all  , where   enumerates the fixed points of   and   is the first fixed point of  . We then have  .

Again, we can see that   for some time: this remains true until the first fixed point   of  , which is the Feferman–Schütte ordinal. Thus,   is the Feferman–Schütte ordinal.

Beyond the Feferman–Schütte ordinal

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We have   for all   where   is the next fixed point of  . So, if   enumerates the fixed points in question (which can also be noted   using the many-valued Veblen functions) we have  , until the first fixed point   of the   itself, which will be   (and the first fixed point   of the   functions will be  ). In this manner:

  •   is the Ackermann ordinal (the range of the notation   defined predicatively),
  •   is the "small" Veblen ordinal (the range of the notations   predicatively using finitely many variables),
  •   is the "large" Veblen ordinal (the range of the notations   predicatively using transfinitely-but-predicatively-many variables),
  • the limit   of  ,  ,  , etc., is the Bachmann–Howard ordinal: after this our function   is constant, and we can go no further with the definition we have given.

Ordinal notations up to the Bachmann–Howard ordinal

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We now explain more systematically how the   function defines notations for ordinals up to the Bachmann–Howard ordinal.

A note about base representations

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Recall that if   is an ordinal which is a power of   (for example   itself, or  , or  ), any ordinal   can be uniquely expressed in the form  , where   is a natural number,   are non-zero ordinals less than  , and   are ordinal numbers (we allow  ). This "base   representation" is an obvious generalization of the Cantor normal form (which is the case  ). Of course, it may quite well be that the expression is uninteresting, i.e.,  , but in any other case the   must all be less than  ; it may also be the case that the expression is trivial (i.e.,  , in which case   and  ).

If   is an ordinal less than  , then its base   representation has coefficients   (by definition) and exponents   (because of the assumption  ): hence one can rewrite these exponents in base   and repeat the operation until the process terminates (any decreasing sequence of ordinals is finite). We call the resulting expression the iterated base   representation of   and the various coefficients involved (including as exponents) the pieces of the representation (they are all  ), or, for short, the  -pieces of  .

Some properties of ψ

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  • The function   is non-decreasing and continuous (this is more or less obvious from its definition).
  • If   with   then necessarily  . Indeed, no ordinal   with   can belong to   (otherwise its image by  , which is   would belong to   — impossible); so   is closed by everything under which   is the closure, so they are equal.
  • Any value   taken by   is an  -number (i.e., a fixed point of  ). Indeed, if it were not, then by writing it in Cantor normal form, it could be expressed using sums, products and exponentiation from elements less than it, hence in  , so it would be in  , a contradiction.
  • Lemma: Assume   is an  -number and   an ordinal such that   for all  : then the  -pieces (defined above) of any element of   are less than  . Indeed, let   be the set of ordinals all of whose  -pieces are less than  . Then   is closed under addition, multiplication and exponentiation (because   is an  -number, so ordinals less than it are closed under addition, multiplication and exponentiation). And   also contains every   for   by assumption, and it contains  ,  ,  ,  . So  , which was to be shown.
  • Under the hypothesis of the previous lemma,   (indeed, the lemma shows that  ).
  • Any  -number less than some element in the range of   is itself in the range of   (that is,   omits no  -number). Indeed: if   is an  -number not greater than the range of  , let   be the least upper bound of the   such that  : then by the above we have  , but   would contradict the fact that   is the least upper bound — so  .
  • Whenever  , the set   consists exactly of those ordinals   (less than  ) all of whose  -pieces are less than  . Indeed, we know that all ordinals less than  , hence all ordinals (less than  ) whose  -pieces are less than  , are in  . Conversely, if we assume   for all   (in other words if   is the least possible with  ), the lemma gives the desired property. On the other hand, if   for some  , then we have already remarked   and we can replace   by the least possible with  .

The ordinal notation

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Using the facts above, we can define a (canonical) ordinal notation for every   less than the Bachmann–Howard ordinal. We do this by induction on  .

If   is less than  , we use the iterated Cantor normal form of  . Otherwise, there exists a largest  -number   less or equal to   (this is because the set of  -numbers is closed): if   then by induction we have defined a notation for   and the base   representation of   gives one for  , so we are finished.

It remains to deal with the case where   is an  -number: we have argued that, in this case, we can write   for some (possibly uncountable) ordinal  : let   be the greatest possible such ordinal (which exists since   is continuous). We use the iterated base   representation of  : it remains to show that every piece of this representation is less than   (so we have already defined a notation for it). If this is not the case then, by the properties we have shown,   does not contain  ; but then   (they are closed under the same operations, since the value of   at   can never be taken), so  , contradicting the maximality of  .

Note: Actually, we have defined canonical notations not just for ordinals below the Bachmann–Howard ordinal but also for certain uncountable ordinals, namely those whose  -pieces are less than the Bachmann–Howard ordinal (viz.: write them in iterated base   representation and use the canonical representation for every piece). This canonical notation is used for arguments of the   function (which may be uncountable).

Examples

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For ordinals less than  , the canonical ordinal notation defined coincides with the iterated Cantor normal form (by definition).

For ordinals less than  , the notation coincides with iterated base   notation (the pieces being themselves written in iterated Cantor normal form): e.g.,   will be written  , or, more accurately,  . For ordinals less than  , we similarly write in iterated base   and then write the pieces in iterated base   (and write the pieces of that in iterated Cantor normal form): so   is written  , or, more accurately,  . Thus, up to  , we always use the largest possible  -number base which gives a non-trivial representation.

Beyond this, we may need to express ordinals beyond  : this is always done in iterated  -base, and the pieces themselves need to be expressed using the largest possible  -number base which gives a non-trivial representation.

Note that while   is equal to the Bachmann–Howard ordinal, this is not a "canonical notation" in the sense we have defined (canonical notations are defined only for ordinals less than the Bachmann–Howard ordinal).

Conditions for canonicalness

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The notations thus defined have the property that whenever they nest   functions, the arguments of the "inner"   function are always less than those of the "outer" one (this is a consequence of the fact that the  -pieces of  , where   is the largest possible such that   for some  -number  , are all less than  , as we have shown above). For example,   does not occur as a notation: it is a well-defined expression (and it is equal to   since   is constant between   and  ), but it is not a notation produced by the inductive algorithm we have outlined.

Canonicalness can be checked recursively: an expression is canonical if and only if it is either the iterated Cantor normal form of an ordinal less than  , or an iterated base   representation all of whose pieces are canonical, for some   where   is itself written in iterated base   representation all of whose pieces are canonical and less than  . The order is checked by lexicographic verification at all levels (keeping in mind that   is greater than any expression obtained by  , and for canonical values the greater   always trumps the lesser or even arbitrary sums, products and exponentials of the lesser).

For example,   is a canonical notation for an ordinal which is less than the Feferman–Schütte ordinal: it can be written using the Veblen functions as  .

Concerning the order, one might point out that   (the Feferman–Schütte ordinal) is much more than   (because   is greater than   of anything), and   is itself much more than   (because   is greater than  , so any sum-product-or-exponential expression involving   and smaller value will remain less than  ). In fact,   is already less than  .

Standard sequences for ordinal notations

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To witness the fact that we have defined notations for ordinals below the Bachmann–Howard ordinal (which are all of countable cofinality), we might define standard sequences converging to any one of them (provided it is a limit ordinal, of course). Actually we will define canonical sequences for certain uncountable ordinals, too, namely the uncountable ordinals of countable cofinality (if we are to hope to define a sequence converging to them...) which are representable (that is, all of whose  -pieces are less than the Bachmann–Howard ordinal).

The following rules are more or less obvious, except for the last:

  • First, get rid of the (iterated) base   representations: to define a standard sequence converging to  , where   is either   or   (or  , but see below):
    • if   is zero then   and there is nothing to be done;
    • if   is zero and   is successor, then   is successor and there is nothing to be done;
    • if   is limit, take the standard sequence converging to   and replace   in the expression by the elements of that sequence;
    • if   is successor and   is limit, rewrite the last term   as   and replace the exponent   in the last term by the elements of the fundamental sequence converging to it;
    • if   is successor and   is also, rewrite the last term   as   and replace the last   in this expression by the elements of the fundamental sequence converging to it.
  • If   is  , then take the obvious   as the fundamental sequence for  .
  • If   then take as fundamental sequence for   the sequence  
  • If   then take as fundamental sequence for   the sequence  
  • If   where   is a limit ordinal of countable cofinality, define the standard sequence for   to be obtained by applying   to the standard sequence for   (recall that   is continuous and increasing, here).
  • It remains to handle the case where   with   an ordinal of uncountable cofinality (e.g.,   itself). Obviously it doesn't make sense to define a sequence converging to   in this case; however, what we can define is a sequence converging to some   with countable cofinality and such that   is constant between   and  . This   will be the first fixed point of a certain (continuous and non-decreasing) function  . To find it, apply the same rules (from the base   representation of  ) as to find the canonical sequence of  , except that whenever a sequence converging to   is called for (something which cannot exist), replace the   in question, in the expression of  , by a   (where   is a variable) and perform a repeated iteration (starting from  , say) of the function  : this gives a sequence   tending to  , and the canonical sequence for   is  ,  ,  ... If we let the  th element (starting at  ) of the fundamental sequence for   be denoted as  , then we can state this more clearly using recursion. Using this notation, we can see that   quite easily. We can define the rest of the sequence using recursion:  . (The examples below should make this clearer.)

Here are some examples for the last (and most interesting) case:

  • The canonical sequence for   is:  ,  ,  ... This indeed converges to   after which   is constant until  .
  • The canonical sequence for   is:  ,  ,   This indeed converges to the value of   at   after which   is constant until  .
  • The canonical sequence for   is:   This converges to the value of   at  .
  • The canonical sequence for   is   This converges to the value of   at  .
  • The canonical sequence for   is:   This converges to the value of   at  .
  • The canonical sequence for   is:   This converges to the value of   at  .
  • The canonical sequence for   is:   This converges to the value of   at  .
  • The canonical sequence for   is:  

Here are some examples of the other cases:

  • The canonical sequence for   is:  ,  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ,  ...
  • The canonical sequence for   is:  ,  ,  ... (this is derived from the fundamental sequence for  ).
  • The canonical sequence for   is:  ,  ,  ... (this is derived from the fundamental sequence for  , which was given above).

Even though the Bachmann–Howard ordinal   itself has no canonical notation, it is also useful to define a canonical sequence for it: this is  ,  ,  ...

A terminating process

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Start with any ordinal less than or equal to the Bachmann–Howard ordinal, and repeat the following process so long as it is not zero:

  • if the ordinal is a successor, subtract one (that is, replace it with its predecessor),
  • if it is a limit, replace it by some element of the canonical sequence defined for it.

Then it is true that this process always terminates (as any decreasing sequence of ordinals is finite); however, like (but even more so than for) the hydra game:

  1. it can take a very long time to terminate,
  2. the proof of termination may be out of reach of certain weak systems of arithmetic.

To give some flavor of what the process feels like, here are some steps of it: starting from   (the small Veblen ordinal), we might go down to  , from there down to  , then   then   then   then   then   then   then   and so on. It appears as though the expressions are getting more and more complicated whereas, in fact, the ordinals always decrease.

Concerning the first statement, one could introduce, for any ordinal   less or equal to the Bachmann–Howard ordinal  , the integer function   which counts the number of steps of the process before termination if one always selects the  'th element from the canonical sequence (this function satisfies the identity  ). Then   can be a very fast growing function: already   is essentially  , the function   is comparable with the Ackermann function  , and   is comparable with the Goodstein function. If we instead make a function that satisfies the identity  , so the index of the function increases it is applied, then we create a much faster growing function:   is already comparable to the Goodstein function, and   is comparable to the TREE function.

Concerning the second statement, a precise version is given by ordinal analysis: for example, Kripke–Platek set theory can prove[4] that the process terminates for any given   less than the Bachmann–Howard ordinal, but it cannot do this uniformly, i.e., it cannot prove the termination starting from the Bachmann–Howard ordinal. Some theories like Peano arithmetic are limited by much smaller ordinals (  in the case of Peano arithmetic).

Variations on the example

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Making the function less powerful

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It is instructive (although not exactly useful) to make   less powerful.

If we alter the definition of   above to omit exponentiation from the repertoire from which   is constructed, then we get   (as this is the smallest ordinal which cannot be constructed from  ,   and   using addition and multiplication only), then   and similarly  ,   until we come to a fixed point which is then our  . We then have   and so on until  . Since multiplication of  's is permitted, we can still form   and   and so on, but our construction ends there as there is no way to get at or beyond  : so the range of this weakened system of notation is   (the value of   is the same in our weaker system as in our original system, except that now we cannot go beyond it). This does not even go as far as the Feferman–Schütte ordinal.

If we alter the definition of   yet some more to allow only addition as a primitive for construction, we get   and   and so on until   and still  . This time,   and so on until   and similarly  . But this time we can go no further: since we can only add  's, the range of our system is  .

If we alter the definition even more, to allow nothing except psi, we get  ,  , and so on until  ,  , and  , at which point we can go no further since we cannot do anything with the  's. So the range of this system is only  .

In both cases, we find that the limitation on the weakened   function comes not so much from the operations allowed on the countable ordinals as on the uncountable ordinals we allow ourselves to denote.

Going beyond the Bachmann–Howard ordinal

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We know that   is the Bachmann–Howard ordinal. The reason why   is no larger, with our definitions, is that there is no notation for   (it does not belong to   for any  , it is always the least upper bound of it). One could try to add the   function (or the Veblen functions of so-many-variables) to the allowed primitives beyond addition, multiplication and exponentiation, but that does not get us very far. To create more systematic notations for countable ordinals, we need more systematic notations for uncountable ordinals: we cannot use the   function itself because it only yields countable ordinals (e.g.,   is,  , certainly not  ), so the idea is to mimic its definition as follows:

Let   be the smallest ordinal which cannot be expressed from all countable ordinals and   using sums, products, exponentials, and the   function itself (to previously constructed ordinals less than  ).

Here,   is a new ordinal guaranteed to be greater than all the ordinals which will be constructed using  : again, letting   and   works.

For example,  , and more generally   for all countable ordinals and even beyond (  and  ): this holds up to the first fixed point   of the function   beyond  , which is the limit of  ,   and so forth. Beyond this, we have   and this remains true until  : exactly as was the case for  , we have   and  .

The   function gives us a system of notations (assuming we can somehow write down all countable ordinals!) for the uncountable ordinals below  , which is the limit of  ,   and so forth.

Now we can reinject these notations in the original   function, modified as follows:

  is the smallest ordinal which cannot be expressed from  ,  ,  ,   and   using sums, products, exponentials, the   function, and the   function itself (to previously constructed ordinals less than  ).

This modified function   coincides with the previous one up to (and including)   — which is the Bachmann–Howard ordinal. But now we can get beyond this, and   is   (the next  -number after the Bachmann–Howard ordinal). We have made our system doubly impredicative: to create notations for countable ordinals we use notations for certain ordinals between   and   which are themselves defined using certain ordinals beyond  .


A variation on this scheme, which makes little difference when using just two (or finitely many) collapsing functions, but becomes important for infinitely many of them, is to define

  is the smallest ordinal which cannot be expressed from  ,  ,  ,   and   using sums, products, exponentials, and the   and   function (to previously constructed ordinals less than  ).

i.e., allow the use of   only for arguments less than   itself. With this definition, we must write   instead of   (although it is still also equal to  , of course, but it is now constant until  ). This change is inessential because, intuitively speaking, the   function collapses the nameable ordinals beyond   below the latter so it matters little whether   is invoked directly on the ordinals beyond   or on their image by  . But it makes it possible to define   and   by simultaneous (rather than "downward") induction, and this is important if we are to use infinitely many collapsing functions.

Indeed, there is no reason to stop at two levels: using   new cardinals in this way,  , we get a system essentially equivalent to that introduced by Buchholz,[3] the inessential difference being that since Buchholz uses   ordinals from the start, he does not need to allow multiplication or exponentiation; also, Buchholz does not introduce the numbers   or   in the system as they will also be produced by the   functions: this makes the entire scheme much more elegant and more concise to define, albeit more difficult to understand. This system is also sensibly equivalent to the earlier (and much more difficult to grasp) "ordinal diagrams" of Takeuti[5] and   functions of Feferman: their range is the same ( , which could be called the Takeuti-Feferman–Buchholz ordinal, and which describes the strength of  -comprehension plus bar induction).


A "normal" variant

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Most definitions of ordinal collapsing functions found in the recent literature differ from the ones we have given in one technical but important way which makes them technically more convenient although intuitively less transparent. We now explain this.

The following definition (by induction on  ) is completely equivalent to that of the function   above:

Let   be the set of ordinals generated starting from  ,  ,  ,   and all ordinals less than   by recursively applying the following functions: ordinal addition, multiplication and exponentiation, and the function  . Then   is defined as the smallest ordinal   such that  .

(This is equivalent, because if   is the smallest ordinal not in  , which is how we originally defined  , then it is also the smallest ordinal not in  , and furthermore the properties we described of   imply that no ordinal between   inclusive and   exclusive belongs to  .)

We can now make a change to the definition which makes it subtly different:

Let   be the set of ordinals generated starting from  ,  ,  ,   and all ordinals less than   by recursively applying the following functions: ordinal addition, multiplication and exponentiation, and the function  . Then   is defined as the smallest ordinal   such that   and  .

The first values of   coincide with those of  : namely, for all   where  , we have   because the additional clause   is always satisfied. But at this point the functions start to differ: while the function   gets "stuck" at   for all  , the function   satisfies   because the new condition   imposes  . On the other hand, we still have   (because   for all   so the extra condition does not come in play). Note in particular that  , unlike  , is not monotonic, nor is it continuous.

Despite these changes, the   function also defines a system of ordinal notations up to the Bachmann–Howard ordinal: the notations, and the conditions for canonicity, are slightly different (for example,   for all   less than the common value  ).

Other similar OCFs

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Arai's ψ

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Arai's ψ function is an ordinal collapsing function introduced by Toshiyasu Arai (husband of Noriko H. Arai) in his paper: A simplified ordinal analysis of first-order reflection.   is a collapsing function such that  , where   represents the first uncountable ordinal (it can be replaced by the Church–Kleene ordinal at the cost of extra technical difficulty). Throughout the course of this article,   represents Kripke–Platek set theory for a  -reflecting universe,   is the least  -indescribable cardinal (it may be replaced with the least  -reflecting ordinal at the cost of extra technical difficulty),   is a fixed natural number  , and  .

Suppose   for a   ( )-sentence  . Then, there exists a finite   such that for  ,  . It can also be proven that   proves that each initial segment   is well-founded, and therefore,   is the proof-theoretic ordinal of  . One can then make the following conversions:

  •  , where   is either the least recursively regular ordinal or the least uncountable cardinal,   is Kripke–Platek set theory with infinity and   is the Bachmann–Howard ordinal.
  •  , where   is either the least limit of admissible ordinals or the least limit of infinite cardinals and   is Buchholz's ordinal.
  •  , where   is either the least limit of admissible ordinals or the least limit of infinite cardinals,   is KPi without the collection scheme and   is the Takeuti–Feferman–Buchholz ordinal.
  •  , where   is either the least recursively inaccessible ordinal or the least weakly inaccessible cardinal and   is Kripke–Platek set theory with a recursively inaccessible universe.

Bachmann's ψ

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The first true OCF, Bachmann's   was invented by Heinz Bachmann, somewhat cumbersome as it depends on fundamental sequences for all limit ordinals; and the original definition is complicated. Michael Rathjen has suggested a "recast" of the system, which goes like so:

  • Let   represent an uncountable ordinal such as  ;
  • Then define   as the closure of   under addition,   and   for  .
  •   is the smallest countable ordinal ρ such that  

  is the Bachmann–Howard ordinal, the proof-theoretic ordinal of Kripke–Platek set theory with the axiom of infinity (KP).

Buchholz's ψ

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Buchholz's   is a hierarchy of single-argument functions  , with   occasionally abbreviated as  . This function is likely the most well known out of all OCFs. The definition is so:

  • Define   and   for  .
  • Let   be the set of distinct terms in the Cantor normal form of   (with each term of the form   for  , see Cantor normal form theorem)
  •  
  •  
  •  
  •  

The limit of this system is  , the Takeuti–Feferman–Buchholz ordinal.

Extended Buchholz's ψ

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This OCF is a sophisticated extension of Buchholz's   by mathematician Denis Maksudov. The limit of this system, sometimes called the Extended Buchholz Ordinal, is much greater, equal to   where   denotes the first omega fixed point. The function is defined as follows:

  • Define   and   for  .
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Madore's ψ

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This OCF was the same as the ψ function previously used throughout this article; it is a simpler, more efficient version of Buchholz's ψ function defined by David Madore. Its use in this article lead to widespread use of the function.

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This function was used by Chris Bird, who also invented the next OCF.

Bird's θ

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Chris Bird devised the following shorthand for the extended Veblen function  :

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  •   is abbreviated  

This function is only defined for arguments less than  , and its outputs are limited by the small Veblen ordinal.

Jäger's ψ

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Jäger's ψ is a hierarchy of single-argument ordinal functions ψκ indexed by uncountable regular cardinals κ smaller than the least weakly Mahlo cardinal M0 introduced by German mathematician Gerhard Jäger in 1984. It was developed on the base of Buchholz's approach.

  • If   for some α < κ,  .
  • If   for some α, βκ,  .
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  • For every finite n,   is the smallest set satisfying the following:
    • The sum of any finitely many ordinals in   belongs to  .
    • For any  ,  .
    • For any  ,  .
    • For any ordinal γ and uncountable regular cardinal  ,  .
    • For any   and uncountable regular cardinal  ,  .
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Simplified Jäger's ψ

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This is a sophisticated simplification of Jäger's ψ created by Denis Maksudov. An ordinal is α-weakly inaccessible if it is uncountable, regular and it is a limit of γ-weakly inaccessible cardinals for γ < α. Let I(α, 0) be the first α-weakly inaccessible cardinal, I(α, β + 1) be the first α-weakly inaccessible cardinal after I(α, β) and I(α, β) =   for limit β. Restrict ρ and π to uncountable regular ordinals of the form I(α, 0) or I(α, β + 1). Then,

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Rathjen's Ψ

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Rathjen's Ψ function is based on the least weakly compact cardinal to create large countable ordinals. For a weakly compact cardinal K, the functions  ,  ,  , and   are defined in mutual recursion in the following way:

  • M0 =  , where Lim denotes the class of limit ordinals.
  • For α > 0, Mα is the set   is stationary in  
  •   is the closure of   under addition,  ,   given ξ < K,   given ξ < α, and   given  .
  •  .
  • For  ,  .

Collapsing large cardinals

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As noted in the introduction, the use and definition of ordinal collapsing functions is strongly connected with the theory of ordinal analysis, so the collapse of this or that large cardinal must be mentioned simultaneously with the theory for which it provides a proof-theoretic analysis.

  • Gerhard Jäger and Wolfram Pohlers[6] described the collapse of an inaccessible cardinal to describe the ordinal-theoretic strength of Kripke–Platek set theory augmented by the recursive inaccessibility of the class of ordinals (KPi), which is also proof-theoretically equivalent[1] to  -comprehension plus bar induction. Roughly speaking, this collapse can be obtained by adding the   function itself to the list of constructions to which the   collapsing system applies.
  • Michael Rathjen[7] then described the collapse of a Mahlo cardinal to describe the ordinal-theoretic strength of Kripke–Platek set theory augmented by the recursive Mahloness of the class of ordinals (KPM).
  • Rathjen[8] later described the collapse of a weakly compact cardinal to describe the ordinal-theoretic strength of Kripke–Platek set theory augmented by certain reflection principles (concentrating on the case of  -reflection). Very roughly speaking, this proceeds by introducing the first cardinal   which is  -hyper-Mahlo and adding the   function itself to the collapsing system.
  • In a 2015 paper, Toshyasu Arai has created ordinal collapsing functions   for a vector of ordinals  , which collapse  -indescribable cardinals for  . These are used to carry out ordinal analysis of Kripke–Platek set theory augmented by  -reflection principles. [9]
  • Rathjen has investigated the collapse of yet larger cardinals, with the ultimate goal of achieving an ordinal analysis of  -comprehension (which is proof-theoretically equivalent to the augmentation of Kripke–Platek by  -separation).[10]

Notes

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  1. ^ a b Rathjen, 1995 (Bull. Symbolic Logic)
  2. ^ Kahle, 2002 (Synthese)
  3. ^ a b Buchholz, 1986 (Ann. Pure Appl. Logic)
  4. ^ Rathjen, 2005 (Fischbachau slides)
  5. ^ Takeuti, 1967 (Ann. Math.)
  6. ^ Jäger & Pohlers, 1983 (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber.)
  7. ^ Rathjen, 1991 (Arch. Math. Logic)
  8. ^ Rathjen, 1994 (Ann. Pure Appl. Logic)
  9. ^ T. Arai, A simplified analysis of first-order reflection (2015).
  10. ^ Rathjen, 2005 (Arch. Math. Logic)

References

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  • Arai, Toshiyasu (September 2020). "A simplified ordinal analysis of first-order reflection". The Journal of Symbolic Logic. 85 (3): 1163–1185. arXiv:1907.07611. doi:10.1017/jsl.2020.23. S2CID 118940547.
  • Takeuti, Gaisi (1967). "Consistency proofs of subsystems of classical analysis". Annals of Mathematics. 86 (2): 299–348. doi:10.2307/1970691. JSTOR 1970691.
  • Jäger, Gerhard; Pohlers, Wolfram (1983). "Eine beweistheoretische Untersuchung von ( -CA)+(BI) und verwandter Systeme". Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse Sitzungsberichte. 1982: 1–28.
  • Buchholz, Wilfried (1986). "A New System of Proof-Theoretic Ordinal Functions". Annals of Pure and Applied Logic. 32: 195–207. doi:10.1016/0168-0072(86)90052-7.
  • Rathjen, Michael (1991). "Proof-theoretic analysis of KPM". Archive for Mathematical Logic. 30 (5–6): 377–403. doi:10.1007/BF01621475. S2CID 9376863.
  • Rathjen, Michael (1994). "Proof theory of reflection" (PDF). Annals of Pure and Applied Logic. 68 (2): 181–224. doi:10.1016/0168-0072(94)90074-4.
  • Rathjen, Michael (1995). "Recent Advances in Ordinal Analysis:  -CA and Related Systems". The Bulletin of Symbolic Logic. 1 (4): 468–485. doi:10.2307/421132. JSTOR 421132. S2CID 10648711.
  • Kahle, Reinhard (2002). "Mathematical proof theory in the light of ordinal analysis". Synthese. 133: 237–255. doi:10.1023/A:1020892011851. S2CID 45695465.
  • Rathjen, Michael (2005). "An ordinal analysis of stability". Archive for Mathematical Logic. 44: 1–62. CiteSeerX 10.1.1.15.9786. doi:10.1007/s00153-004-0226-2. S2CID 2686302.
  • Rathjen, Michael (August 2005). "Proof Theory: Part III, Kripke–Platek Set Theory" (PDF). Archived from the original (PDF) on 2007-06-12. Retrieved 2008-04-17.(slides of a talk given at Fischbachau)