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So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily practicing mathematicians) I am curious as to what lessons mathematicians draw from the existence of non-standard models.

Within philosophical circles it seems to be the consensus (or at least a fairly popular view) that what non-standard models show us is something about the limits of formalization. For instance, Haim Gaifman in a lecture delivered to the AMS Special Session Nonstandard Models Of Arithmetic And Set Theory (January 15-16, 2003, Baltimore, Maryland) notes the following:

If set theory is about some domain that includes uncountable sets, then any countable structure that satisfies the formalized theory must count as an unintended model. From the point of view of those who subscribe to the intended interpretation, the existence of such nonstandard models counts as a failure of the formal system to capture the semantics fully.

Now among those who subscribe to this sort of view, they tend to take the failure of categoricity in a first-order theory of Peano Arithmetic to show us that it is a second-order formulation of Peano Arithmetic which is needed. I've always taken this to result from a view that it is the semantic, rather than the syntactic, side of mathematical theorizing which holds some primacy. Often this is coupled with a view taken from Hilbert that mathematical theories (at least those that seem to have an intended interpretation) implicitly define some concept or some structure which various isomorphic models satisfy. On this sort of view, it is the concept which is of primary interest and the deductive systems are a means to discover a bit about (but generally, for reasons of incompleteness, only a bit about) what, for lack of a better word, you might call the nature of this concept.

TL;DR What lessons have mathematicians drawn from the existence of non-standard models?

Mikhail Katz
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Dennis
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    Why draw lessons from them at all? They’re interesting objects of study in their own right, and sometimes useful tools. Apart from that, I agree (vehemently, if not violently!) with @André’s comment. – Brian M. Scott Apr 08 '13 at 05:10
  • @BrianM.Scott Deleted then. It seems that it doesn't add anything and merely distracts from what I am actually interested in. Many apologies for stating what seems now to be certainly a naive and misinformed impression. – Dennis Apr 08 '13 at 05:22
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    The question about non-standard models is interesting. For most mathematicians, they are of no particular interest, mathematics is very diverse. For some, they tell us more about the world, just like continuous functions with "pathological" properties do. – André Nicolas Apr 08 '13 at 05:25
  • @AndréNicolas Thanks for the input. It is an expansion on this sentiment--- "they tell us more about the world" ---that I am interested in, if that helps clarify things. (Note: Some comments above deleted which are no longer relevant to the revised question). – Dennis Apr 08 '13 at 05:28
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    In my humble opinion, mathematicians are better at using tools than drawing lessons. Non-standard models are useful as tool, and one might even risk a statement that tools is precisely what they are. For instance, if you want to prove a first-order statement about "the" reals, it might be useful to use a non-standard model that offers infinitesimals (and then shift back to whatever model is "standard"). Generally, I would say that existence of non-standard models tells us that there are a lot of models that approximate the standard one pretty well. – Jakub Konieczny Apr 08 '13 at 07:48
  • @QiaochuYuan I thought I gave one in the body of my question, sorry for the lack of clarity. One lesson you might draw is something like: "Non-standard models show us that (certain) first-order theories have a crucial limitation: they can't distinguish between models that capture what we intend to say about the natural numbers (for instance) and models which, in some sense, have features that strike us as 'unintuitive' or at least, unintended." You might extend this idea to saying that, really, a second-order formulation of Peano Arithmetic is best (since it avoids this limitation). – Dennis Apr 08 '13 at 15:00
  • (For those who are really interested in examples of lessons, you might skim the linked pdf. Gaifman explains these ideas far better than I can.) – Dennis Apr 08 '13 at 15:12
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    @Dennis: Amusingly, I would draw the opposite lesson: when we learn two things are indistinguishable, we shouldn't be trying to distinguish them! –  Feb 01 '14 at 01:55

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As you point out, Gaifman wrote:

From the point of view of those who subscribe to the intended interpretation, the existence of such nonstandard models counts as a failure of the formal system to capture the semantics fully.

Note that this is not merely about first-order logic but a broader observation.

To answer your question, some mathematicians have drawn the lesson that the entity called the intended model does not exist and that belief in such an entity is an obstacle to progress.

Mikhail Katz
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  • Thanks for your contribution. I'm sympathetic to such a claim, and find partial expression of it in some of Joel David Hamkins's work on the set-theoretic multiverse and his arguments that satisfaction is not absolute. Do you know of other examples in print? – Dennis Aug 21 '17 at 16:38
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    Yes indeed ;-) @Dennis – Mikhail Katz Aug 21 '17 at 16:43
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    Let me observe - and this is meant positively - that one need not adopt the viewpoint of the last few words of this answer completely for this answer to be useful. In particular, a weaker stance (which I personally hold) is that disbelief can be helpful to mathematical progress, without making any negative claim about belief. (Also note that genuine disbelief in standard models raises its own problems. For example, what do we mean when we say that a statement about the natural numbers is true? This isn't a stupid question - my point is just that it is forcefully raised by this answer.) – Noah Schweber Aug 21 '17 at 16:47
  • @NoahSchweber good point! I take it the crucial point is that a dogmatic belief that we have fixed on the standard model, and dismissal of non-standard models in light of this belief, is "mathematically bad" since it preempts potentially fruitful investigation of such models. – Dennis Aug 21 '17 at 16:56
  • When the Western Roman Empire began to disintegrate, Augustine of Hippo developed a doctrine disassociating the survival of the empire from the survival of civilisation. @NoahSchweber – Mikhail Katz Aug 21 '17 at 16:59
  • @MikhailKatz . . . Ok? I'm not really sure what your point is, there. – Noah Schweber Aug 21 '17 at 17:30
  • @Noah, there is no contradiction between statements about natural numbers being meaningful, on the one hand, and questioning the assumptions behind "intended models", on the other. It is possible that some people who are used to making such assumptions think that such assumptions are necessary in order to assign meaning to statements about natural numbers, but the onus is on them. – Mikhail Katz Aug 22 '17 at 08:33
  • @MikhailKatz I didn't say that there was a contradiction - I said that the question "what do we mean when we say that a statement about the natural numbers is true?" is hard to answer if we don't have an intended model in mind. (How do we interpret quantifiers, exactly? As ranging over what?) For instance, there are those who argue, based on the claimed inherent vagueness/nonexistence of the"intended" $\mathbb{N}$, that while low-quantifier statements are meaningful, higher-quantifier ones may be neither true nor false. – Noah Schweber Aug 24 '17 at 02:04
  • @Noah, do you have a source discussing the issue of interpreting statements about arithmetic and the relevance of the intended model for this? As it stands the connection seems dubious to me. I would like to relate to this seriously but if I could see a case made in a longer discussion it might help. – Mikhail Katz Aug 24 '17 at 11:53
  • To take a concrete example, there are arithmetic statements that are true but not provable, according to Goedel. This would seem to prove your point but as you know Goedel's formulation was a function of his philosophical stance. One can just as well formulate the result as asserting the existence of assertions that can be neither proved nor disproved, so this example actually proves my point. Note that the mathematician doing the disagreement with Goedel here is not your humble servant but rather Abraham Robinson, who had a detailed exchange in which he disagreed with Goedel's views on NSA. – Mikhail Katz Aug 25 '17 at 07:22
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I do not know about any lessons, but the conclusions are pretty much agreed. As you said, it implies that first order formal systems are not strong enough to model a single "structure" (or model). Because of Godel incompleteness theorem you will always have additional models than the one you intend to formalize. On the other hand, almost everybody agrees that going to full second order logic (and fixing the semantics) is not a solution either, because second order systems have a lot of problems of their own (there is a trade off), for instance, Quine pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not logic, properly speaking. In the end, what this means depends on you philosophical point of view. If you are a Platonist, it means that no formal system will be capable of proving all the truths of the structure of your choice (for instance, N). If you are a nominalist, it means that, at least in practice, first order formals systems describe an infinite number of "structures" (or models) (you would need a non-recursive enumerable infinite number of axioms to pinpoint a single model).

Wolphram jonny
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    Actually the GIT and plurality of models results are not the same thing. Elementary Euclidean geometry is a complete theory and yet it also has infinitely many models. Even if you take the non-recursive theory of natural numbers, it still has infinitely many models. The problem is first order logic cannot discriminate amongst the infinite. – The_Sympathizer May 19 '15 at 11:50