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This question might sound facetious, but it is a genuine question which I am very much interested in. I apologize in advance if it is too conceptual or philosophical, but I'm optimistic that I might gain some mathematical insight from an answer.

There has been a long standing interest ever since Godel to add new and "true" axioms to set theory. I take it to be definitional that the point of such a program is to eliminate/reduce "non-standard" models of set theory, where a model's non-standardness is judged either by its fit to our intuitive concept of "set" and/or "size" or by some other metaphysical or aesthetic standard. It seems to be the case that a rather trivial part of our conception of the set theoretic universe is that there exist no sets that are models of all set-theoretic truth. That is, every model of set theoretic truth (which, like everything, is a set) will be non-standard in all sorts of ways. It will be absolutely tiny since it is a set rather than a proper class, it won't have all the "real" cardinals, or the "real" membership relation (sometimes), etc. So, my case rests on the following claim:

(1) Every model of set theory (which is a set) will be non-standard according to our conception of the entire set-theoretic universe.

However, once (1) is granted, doesn't it trivially follow that set-theoretic truth (where truth is determined by our conception rather than the axioms) should be inconsistent, since being inconsistent is equivalent to not having any models? If so, doesn't this have serious implications for math and\or philosophy? (i.e. if our very conception of set is inconsistent wouldn't this undermine the "realist" program of finding axioms that capture this conception?)

Taro
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  • This might be relevant, or even a duplicate: http://math.stackexchange.com/questions/370240/does-mathsfzfc-neg-mathrmcon-mathsfzfc-suffice-as-a-foundations-of/ – Asaf Karagila Nov 05 '14 at 16:14
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    What's the point in downvoting an obviously carefully posed question?! – Hanno Nov 05 '14 at 16:16
  • @AsafKaragila That question doesn't give any motivation for why the inconsistency of ZFC should be true. Also, I'm not advocating that ZFC is inconsistent; I'm only saying that the theory embodied in our conception should be inconsistent. The theory embodied by our conception is stronger than ZFC (I think). So, it might well be that ZFC is consistent even though the theory embodied by our conception is not. – Taro Nov 05 '14 at 16:17
  • @Hanno Thanks for the encouragement! I've tried my best to make the question substantive and well-posed, although I expected it might not be well-received. – Taro Nov 05 '14 at 16:19
  • @David: Either the answer is trivial "We use it as a meta-theory, why would you want to assume that your meta-theory is inconsistent?" or that the answer is more mathematical that you don't want the universe of sets to disagree with our internal notion of integers, which seems to be a particularly striking reason -- which is discussed in my answer there. – Asaf Karagila Nov 05 '14 at 16:24
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    The key issue is with "our very conception of set"; the Early Development of Set Theory shows us that there are different "pre-mathematical" concepts of "set" in place, like : set as "property" and set as "collection". The first attempt (Frege, Cantor) to "elucidate" those concepts give rise to problems. The mathematical theory of sets has successfully faced with those problems but (up to now) has not been able to "capture" all the pre-mathematical intution (?) about sets. Mathematics (usually) does not like inconsistency. – Mauro ALLEGRANZA Nov 05 '14 at 16:26
  • @AsafKaragila Do you agree or disagree with the my question (1)? If yes, do you agree or disagree that (1) implies that our concept of set is inconsistent? As for your remarks, perhaps I should have titled my question: "Why isn't our concept of set inconsistent?" Again, I'm all for ZFC being consistent. – Taro Nov 05 '14 at 16:28
  • @DavidBuiles By (1) do you mean that every model fails to satisfy some true claim in $\mathcal L_\in$? –  Nov 05 '14 at 16:43
  • Models are non-standard when "referred to" the standard model, like $\mathbb N$: in this case (I hope ...) we have a clear intuition of natural numbers and their properties. Thus, we call non-standard any model of the first-order arithmetic taht is not isomorphic to it. If our intuition of sets is a "mix" of different concepts with different properties, some of them not "compatible", it seems to me that is hard to speak of "the standard model". – Mauro ALLEGRANZA Nov 05 '14 at 16:44
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    @Hanno I didn't downvote, but it's very hard to find the actual question in there, and once you find it it's just a question about basic definitions (basically: "what does it mean to be a model of a theory" -- see WillO's answer). – Najib Idrissi Nov 05 '14 at 16:46
  • @David: Yes, maybe that would be a better title. The answer is simple "Maybe it is, we don't know." you seem to mix some Platonist approach with a syntactic-semantic approach. Not to mention that the term "standard model of set theory" already exists, and I can't be sure whether or not you are talking about it or not (for example standard models satisfying $\sf CH$ and its negation both exist or don't exist at the same time). – Asaf Karagila Nov 05 '14 at 16:55
  • @AsafKaragila I meant to use the term "non-standard" just as people use it when talking about models of Peano Arithmetic. My apologies if it was confusing. – Taro Nov 05 '14 at 16:57
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    As @Mauro pointed out, there's no "one standard model" of set theory. The term simply means a model which agrees with the universe on the notion of membership (or isomorphic to one which does). This means that if there is one standard model, there are plenty of them which satisfy different theories. You seem to mean that as "The True Model" in some Platonist meaning of the word, like we treat the natural numbers. But even that notion is a bit fickle, since different models of set theory have different theories of arithmetic, so they have different "standard models of arithmetic" [...] – Asaf Karagila Nov 05 '14 at 16:59
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    [...] Of course, from the point of view of a model of set theory, the standard model of arithmetic is unique and deserves to be called "the", but if you take a less-Platonist approach and more of a multiverse-based approach, then you get to switch from one model of set theory to the other, and those might not agree on their integers, and so they might not agree on what a "standard model of arithmetic" is and what is its theory. This is sort of a philosophical relativism, which allows you to change your position depending on your model. But it's exactly why we assume $\sf ZFC$ is consistent. – Asaf Karagila Nov 05 '14 at 17:01
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    Since if it is not consistent, either we are stupid enough to try and write proofs from an inconsistent theory, and still fail to write them properly; or we assume that in the meta-theory of set theory (taking a formalist notion for a moment), the integers are different than the integers of the universe of sets, which means that the way we use set theory to found the rest of the mathematics is... off base. Or, if you want to consider a Platonist view, then simply we're just doing it wrong, what else do you propose we do instead? – Asaf Karagila Nov 05 '14 at 17:03
  • @AsafKaragila For your first two comments, I think I agree with everything you say, but I do want to point out that I don't think the argument taken above is "too Platonist" - i.e. it doesn't assume that our conception of sets fixes a unique V. Even if our conception is vague, which it probably is, I would say that any precisification of our conception of set will satisfy (1). And that is all that is needed. – Taro Nov 05 '14 at 17:04
  • David, when you say that there is a "true statement", then you essentially say that there is some absolute meaning to "true", because truth is a very very very very very relative meaning in mathematics. It is defined in a particular way about a structure. If you say "true" without mentioning the structure, then you mean to say that there is some abstract meaning to the notion of truth. This seems, at least without further explanation, to be a Platonist approach to the issue. If it's not, then you don't explain it very well, I think. – Asaf Karagila Nov 05 '14 at 17:09
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    @AsafKaragila I don't think I ever denied that this was a Platonist approach to truth. (Above, I just said it wasn't too optimistically Platonist that our conception uniquely fixed V). I'm just using a conception of truth that every ordinary non-mathematician would subscribe to when talking about arithmetic, and I'm applying that conception of truth to set theory. – Taro Nov 05 '14 at 17:15

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You are, I think, failing to distinguish between "the collection of all true statements about sets" and "the collection of all those true statements about sets that can be expressed in a given formal language".

What we know from the completeness theorem, etc, is that if the latter collection is consistent, then it has a model. It does not follow that if the former collection is consistent, then it has a model.

(This grants for the sake of argument that we can make sense of the notion of "all true statements about sets" --- whether we can or not is a separate issue from the above).

WillO
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  • I do not agree with your first paragraph : "the collection of all those true statements about ... " by our "natural" conception of truth is by definition consistent. We do not "like" a world where we have two statements $A$ and $\lnot A$ such that both are asserted as true. What the Compl Th says is that "if a collection of sentences (that can be expressed in a given formal language) is consistent, then it has a model". – Mauro ALLEGRANZA Nov 05 '14 at 16:54
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    @MauroALLEGRANZA: With what, exactly, do you disagree? I claim that a) The Completeness Theorem applies to collections of sentences expressible in a formal language, and b) the OP attempts to apply it to a collection of sentences that is not expressible in a formal language. Do you disagree with a), b), or both? – WillO Nov 05 '14 at 17:02
  • @WillO So it it seems you are granting (1) (or at least for the sake of argument), but saying nonetheless that that does not imply our conception of sets is inconsistent since it might be the case that those truths about sets that make (1) true are in principle impossible to formalize. I think I agree with you. It all hinges on two questions (A) Are the truths that make (1) true in principle impossible to formalize? and (B) Can there be a conception that satisfies (1) yet nonetheless be consistent? – Taro Nov 05 '14 at 17:09
  • @MauroALLEGRANZA: ""collection of true sentence" has always a model, irerspective of formal logic and Completeness Th: if it is true it must be "true of something"." I think we are not communicating effectively because we are using words differently. I am using the word "model" in the sense of model theory, so that in particular, a model must be a set, not just "something". – WillO Nov 05 '14 at 17:12
  • So (B) would be asking is there some other method (not relying on the Completeness theorem) to prove that any mathematical conception that satisfies (1) is inconsistent? I don't know the answers to (A) or (B), but it does not seem obvious to me that the answers to them are both "Yes". I agree however that your observation takes out much of the force of the argument I sketched above, so thanks for that. – Taro Nov 05 '14 at 17:13
  • @DavidBuiles: Yes, I think you've pretty much got it. – WillO Nov 05 '14 at 17:14
  • On reflection, however, I tend to agree with the comments of @MauroALLEGRANZA – Taro Nov 05 '14 at 17:22