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?)

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    $\begingroup$ What's the point in downvoting an obviously carefully posed question?! $\endgroup$ – Hanno Nov 5 '14 at 16:16
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    $\begingroup$ 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. $\endgroup$ – Mauro ALLEGRANZA Nov 5 '14 at 16:26
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    $\begingroup$ @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). $\endgroup$ – Najib Idrissi Nov 5 '14 at 16:46
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    $\begingroup$ 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" [...] $\endgroup$ – Asaf Karagila Nov 5 '14 at 16:59
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    $\begingroup$ [...] 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. $\endgroup$ – Asaf Karagila Nov 5 '14 at 17:01

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).

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  • $\begingroup$ 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". $\endgroup$ – Mauro ALLEGRANZA Nov 5 '14 at 16:54
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    $\begingroup$ @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? $\endgroup$ – WillO Nov 5 '14 at 17:02
  • $\begingroup$ @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? $\endgroup$ – Taro Nov 5 '14 at 17:09
  • $\begingroup$ @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". $\endgroup$ – WillO Nov 5 '14 at 17:12
  • $\begingroup$ 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. $\endgroup$ – Taro Nov 5 '14 at 17:13

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