Talk:Skolem's paradox/GA1

GA Review

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Nominator: Pagliaccious (talk · contribs) 04:15, 22 August 2024 (UTC)

Reviewer: David Eppstein (talk · contribs) 23:23, 26 August 2024 (UTC)

First read-through: meaning and completeness

First sentence: should "first-order model of set theory" maybe be "model of first-order set theory"? And how can a model prove anything? It is mathematicians who prove things, not models. Or by extension you could maybe say that an axiomatization proves something (its theorems) but it is still not the model that is the subject of "prove".

Somewhere in the article, and maybe also in the lead, it should be clarified what a model is. I am familiar with them but I think we should not expect all readers to be. I think the punch line of the paradox should also be made more explicit: because an uncountable set exists, such a set is an element of the model, despite the model being (externally) countable.

In the lead "Formally, Skolem's paradox is that every countable axiomatization of set theory in first-order logic, if it is consistent, has a model that is countable": isn't this really just the Löwenheim–Skolem theorem, not the paradox itself? What meaning does "Formally" add? And in the next sentence, about proving existence of uncountable sets, maybe Cantor should be briefly credited?

I suspect that the last line of the lead 'More recently, the paper "Models and Reality"...' is intended to be a summary of the last paragraph of the "Later opinions" section, on Putnam and reactions to Putnam, but this is unclear because that paragraph never mentions "Models and Reality" by name.

You have the dates of Cantor's uncountability theory and the Löwenheim–Skolem theorem, but not of Skolem's observation that they are (philosophically at least) contradictory. Can it be dated?

Doesn't the second paragraph of "The result and its implications" lead to a different paradox? The paradox as stated earlier is "this model is countable but yet it (its domain) contains an element that (in the theory of the model) is uncountable". But the second paragraph doesn't talk about an element that is uncountable in the theory of the model; it talks about a set that is not in the model (the set of all subsets of the model). And I don't see how the part about "meaning we cannot put each set of the model B in relation with some natural number" follows. It somehow seems to be assuming that each subset of the model is a set of the model; why? This paragraph seems to be a mixed-up combination of the Skolem paradox and a different paradox: if B is a countable model, then there exists a set of all subsets of B (with the cardinality of the continuum) but we know that "the set of all sets" cannot actually exist. (The resolution to this paradox being that this set of all subsets of B is not an element of B so it does not belong to the model.)

In the third paragraph of the same section "Skolem resolved the paradox by concluding that the existence of such a set cannot be proven in a countable model". What do you mean by proving something in a model? I think the resolution is that, if U is an element of B that models an uncountable set, and C is an element of B that models a countable set, then there does not exist an element phi of B that models a bijection from U to C. It's not merely that there is no proof element in B that phi models such a bijection, but that this bijection is not part of the model at all.

How does the "formally" of the fourth paragraph introduce a different level of formality than the "formally" of the second paragraph? Especially as both paragraphs appear to consist of informal prose rather than formalized logical deductions? In any case I think this fourth paragraph much more closely describes the paradox and its resolution than the second. One quibble: "There are two special elements of M; they are": maybe instead of "they are", more accurate would be "they model"?

In the fifth paragraph, we again have this confusing issue of whether a model contains an element that models a bijection, or whether the bijection can be proven to exist. Why not "relative to one model, no enumerating function puts some set into correspondence with the natural numbers, but relative to another model, this correspondence may exist"? What does provability have to do with it, except for the side point that if an object is actually proven to exist (in the theory) then it must exist (in the model that models the theory)?

In the "Reception" section, how exactly does "Skolem's result" (by which I imagine is intended the paradox, not the Lowenheim–Skolem theorem) prove that first-order set theory cannot be categorical? I mean, it cannot be categorical, but how does that follow from the paradox?

The van Dalen & Ebbinghaus reference at the end of the Reception paper discusses a 1937 paper of Zermelo which, van Dalen & Ebbinghaus state, was intended to refute the Skolem paradox. Why is this work not discussed in more detail in this section? The sentence "It is now known that Skolem's paradox is unique to first-order logic; if set theory is studied using higher-order logic with full semantics, then it does not have any countable models, due to the semantics being used." appears to be referenced to van Dalen & Ebbinghaus p145 but that page says nothing about higher-order logic not having countable models; can this be sourced properly?

A minor formatting note: the |30em in the reflist of the References section is no longer needed (reflists are put into columns by default), but Template:Reflist suggests using 20em instead for articles with shortened footnotes, as used here. When I view this article on my laptop I get only one tall column of references, so a smaller number like 20em would make it more likely that a compact two-column format could be used.

I considered the possibility of discussing the proofs of both Lowenheim-Skolem and Cantor's uncountability theorem, but ultimately decided against requesting that. I think it would be too duplicative of material that belongs better on the articles on those theorems. The only important part here is merely that both are proofs in the first-order theory and therefore must be true of any model of the theory.

The illustrations are more decorative than informative, but they are properly licensed and have informative and relevant captions; I think they're ok, and it's hard to imagine what else might be used as an illustration for this article.

I think that covers WP:GACR criteria 1b, 3, and 6. Criteria 4 and 5 are unlikely to be problematic, but I still need to do another read-through (another day soon, most likely) for low-level copyediting (1a) and for cross-checking the references against the material they reference (2; touched on briefly above but not thoroughly checked).

—David Eppstein (talk) 06:59, 27 August 2024 (UTC)

Hello David Eppstein. Thank you for taking the time to write such a thorough review. I think that I've addressed all of your comments:

• Throughout the article, I've done my best to remove "prove" or "proven" where possible, replacing it with "satisfied" whenever necessary, as you described at several points.
• I've clarified what a model is in the lead. If you think that the article would be improved by a longer explanation, I would gladly move it to the "Background" section and expand it.
• I've removed all the "Formally" introductory phrases, mentioned Cantor in the lead, and fixed the mention of Putnam
• Added date of Skolem's paper
• Removed the second paragraph of the Result section
• Concerning the "first-order set theory cannot be categorical" claim: this article is a bit of a forgotten battleground, as you might tell from the talk page. This claim is an artifact from this era of the page, which I missed when sourcing and cleaning up the existing prose. It seems to only be mentioned on this course webpage, but nothing published. This source was a topic of much discussion on the talk page. I'm happy to remove it from the article.
• I've sourced the "Skolem's paradox is unique to first-order logic" statement properly. The van Dalen & Ebbinghaus only sourced the following sentence, on Zermelo's 1937 paper.
• Concerning this 1937 paper of Zermelo: in fact, it is not so much a paper as a handwritten rough draft. It appears in English on pages 155-156 of the van Dalen and Ebbinghaus paper. From what I understand, this is Zermelo's last attempt at an attack on "finitism" by means of attacking the paradox, and it was left unfinished, unpublished, and uncirculated until the van Dalen and Ebbinghaus paper. It's interesting within the context of the history of the paradox, but I don't know if it's very important to a broader understanding of the paradox's history, as it was left unpublished. I've added a short sentence clarifying that the refutation was unfinished.
• Fixed the ref formatting
• For the images, although you said that they seemed to be fine, I've decided to remove all but Skolem's portrait. I agree that they're only decorative, and Zermelo's and Putnam's pictures were a bit irrelevant.

Kind regards, Pagliaccious (talk) 14:45, 28 August 2024 (UTC)