What exactly is the distinction between a theory and model in model theory if models are themselves constructed in axiomatic theories?

Question

My understanding is that model theory requires a distinction between a logical theory and a structure to interpret the statements of the theory.

However, every piece of mathematics including every mathematical structure relies on some axiomatic system (formalized or naive). So any model itself is going to rest upon some theory. With that in mind, what does it even mean to make the theory/model distinction?

For an example, we can consider Tarski's axioms to generate a theory $T$. Then we can consider the plane $\mathbb{R}^{2}$ with appropriate notions of lines, points, etc. as a model of $T$. But $\mathbb{R}^{2}$ is a construction borne out from ZFC set theory. Hence the truth/falsity of statements about $\mathbb{R}^{2}$ is itself a logical procedure.

Ultimately, are we just "modeling" a theory with another theory? If that's the case, then given a theory $T$ why can't I just use $T$ with the additional axiom [there exists an object satisfying $T$]? Is there a rigorous definition of model that would clear up my confusion?

Answer 1

Caveat: I am not really educated in mathematical philosophy, nor do I deal extensively in mathematical foundations, so the below is just an account of my perspective as a working model theorist.

It's also a bit of a rant, but the question is sufficiently soft that I think it is unavoidable if you want to provide a reasonable answer.

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A precise answer to this question will heavily depend on your philosophical stance (regarding the nature and/or existence of mathematical objects), and I hardly consider myself qualified to give an exposition of the possible answers and their intricacies. Further, it depends on the exact foundation for mathematics you subscribe to. But I will try to give a rough answer of how you might think of it as a working mathematician, without worrying too much about philosophy.

The typical "lazy" attitude to resolve this sort of question (well, besides ignoring it entirely) is to take for granted the platonic ideal of absolute existence of mathematical objects as elements in a fixed universe $V$, which, furthermore, satisfies the axioms of ZFC (or some other fixed set theory you think is sufficiently close to "truth").

Now, this may not be entirely satisfying, because you might ask what it means for $V$ to satisfy the axioms of ZFC. After all, to define what it means for $V$ to satisfy the axioms, you need $V$ to, well, exist, and if $V$ is to satisfy ZFC, then $V\notin V$, and if all mathematical objects are to be members of $V$, it follows that $V$ is not a mathematical object. Thus, asking whether it satisfies the axioms of ZFC is nonsense.

To resolve this, you can say that, well, all mathematical objects are elements of $V$, but $V$ exists as a metamathematical object, and as such, it is a model of ZFC. But this raises further question: where do these metamathematical objects come from, and again, what does satisfaction mean?

One possible answer is to think that we have an even "bigger" model of set theory. At first, it may seem that this does not help us, since we're back where we came from. However, note that this "bigger" model does not need to satisfy all of ZFC. It only needs to satisfy enough for us to be able to check whether $V$ satisfies the axioms of ZFC, and to determine their consequences.

In fact, due to the finitary nature of first order logic, we don't really need this bigger model of set theory: the axioms of ZFC are recursively denumerable (under Gödel numbering), and the notion of proof in FOL is inherently recursive. Thus, in order to describe properties of $V$ (and their consequences), it is actually enough to know a little bit of arithmetic. How much exactly - that is what reverse mathematics deals with, more or less. In this view, we can abandon the idea of the absolute existence of $V$ (and other mathematical objects besides the natural numbers) - we can think of them as sort of "syntactic sugar" for the consequences of the axioms of set theory, at least as far as the consequences of the axioms are concerned.

Alternately, instead of going straight to arithmetic, we can go to something else, like type theory, or some set theory weaker (and thus more obviously intuitively "real") than ZFC.

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In any event, unless we commit to some sort of infinite descent, you reach a point where you have to take something for granted, like some basic properties of the arithmetic of the natural numbers. In the end, the most basic foundation comes from our intuition about logic and arithmetic. Depending on your philosphical inclinations, you can believe that this intuition is artificial, as is the mathematical reality, or that this intuition is correct, and the mathematical reality - genuine. Or something else entirely.

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In my view, to do mathematics, you don't need to assume any of that. It is sometimes useful, but actually (in my view) nonsensical to say that natural numbers, for example, are sets (even though they are modelled quite nicely by some sets, and it is sometimes useful to identify them with those sets). Rather, the idea of the natural numbers, like the idea of the real numbers, is largely independent from the idea of a set. When doing mathematics, it is helpful to "model" (in a more loose sense) these ideas as sets in order to be able to do with them some things we expect to be able to do with sets (like taking a power set, or fixing a wellordering), but we don't really need to think that these things are sets in a model of ZFC, only that they are sufficiently similar to such elements. This is why it is good that we can construct e.g. the reals as Dedekind cuts of rationals - this shows that there is a set that behaves more or less like the real numbers should.

Bottom line: I believe that most (I would like to say all, but there probably are some oddballs) mathematicians almost never think of real numbers as equivalence classes of Cauchy sequences or Dedekind cuts, nor does he, in his mind, equate the number $0$ with $\emptyset$ - because usually, it is not helpful, and if you think a bit, it is not really true. The things we study in mathematics are not usually built by putting sets inside other sets, but rather, by piling ideas on top of other ideas. The question of foundations may be intriguing, but (for a working mathematician) typically irrelevant, and, to some degree, hopeless.

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The most baffling thing about this all, really, is that math works surpisingly well. Perhaps because the intuition is based on the physical reality and brains evolved to interact with it.

Answer 2

I've had much the same questions/confusions for a long time. I am also not an expert in this area, but hopefully by posting an additional attempted answer to this question could help. If in no other way than to prompt someone better qualified to give a better answer.

Ultimately, are we just "modeling" a theory with another theory

yes, or at least this is ultimately the conclusion I also came to.

More specifically, it'ts worth noting that the theory being modelled is the "object theory", and the theory we are using "to do modelling" is the "metatheory". Cf. Rob Arthan's comment.

If that's the case, then given a theory why can't I just use with the additional axiom [there exists an object satisfying ]?

This is a good question. There seem to be at least two issues.

(1) that axiom would most likely not be formalizable in T. Considering T as just a formal string of symbols without any inherent meaning, heuristically speaking we might expect that those symbols can not have any meaning except in the context of an interpreting metatheory. I.e. the idea being that "truth" or "meaning" is always relative, never absolute. (Even when it "seems absolute", it's reasonable to argue that this is only "relativization with respect to an identical copy, cf. below".)

So at best an axiom could only formalize something like "an object exists satisfying something that syntactically looks the same as T", but heuristically speaking no T could ever be able to directly refer to itself. Cf. the use of "name" notation $\lceil{x}\rceil$ in discussions of the liar paradox and/or the "nondefinability of truth" (mentioned more below).

Again c.f. the object language/theory vs metalanguage/theory distinction Rob Arthan mentioned in the comments. One good source to start is discussion of the Liar paradox, cf. e.g. this artitcle that's decently syntactical / formal.

(2) trying to get around that claimed restriction, e.g. by using the formal language to interpret (an identical copy of) itself (which is totally possible), i.e. two copies of the formal language, one as the object language, one as the metalanguage, turns out to "not be possible" in a specific sense.

Heuristically the idea seems to be that, for any language "complex" enough to be interesting (such as the natural numbers), any such attempt to define "truth" (in particular the assertion that "an object exists satisfying T") can not work. If the language/theory T is "complex enough to be interesting", then it's also "not powerful enough to define objects satisfying T".

This is Tarski's theorem on the nondefinability/undefinablity of truth (cf. e.g. here or here). It is related to Godel's incompleteness theorem.

(Warning: the "completeness" referred to in Godel's "incompleteness theorem" is a completely different kind of "completeness" than that referred to in Godel's "completeness" theorem.)

Heuristically speaking this result explains the commonly seen pattern of using e.g. ZF(C) to exhibit models of (interpret) the theories of Peano arithmetic (natural numbers) or of GST (finite sets) that witness their consistency, or using a theory of ZFC+large cardinals to exhibit models of ZFC that witness ZFC's consistency.

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There is a lot about this question that is worthy of further comment and discussion, as well as some of the points brought up in the comments and other answer. E.g. note that all of the above only requires formalizing at most computable functions between finite sets of strings, which can be done in very weak logical theories whose "physical existence" is easily "empirically witnessed" e.g. via computer implementations. So it requires very little "ontological commitment", at least nothing an ultrafinitist (cf. here or here) would find objectionable.

However, I'm tired and it's late at night for me right now though, and the above is already substantially long, so I'll cut this answer off prematurely here. But hopefully the attempt to directly address the last paragraph of your question is at least somewhat helpful.

P.S. You might find this expository paper by Yanofsky fun/interesting (arXiv version) (jstor preview) because it relates the above ideas about the Liar paradox, nondefinability of truth theorem, Godel's incompleteness theorem, to Cantor's diagonalization argument, the halting problem, and others. Someone even made a YouTube video about it. But it's not necessarily useful for understanding the object theory ("syntax") vs. metatheory ("semantics" but can be formalized as "meta-syntax") distinction that implicitly needs to be used to understand mathematical logic, and which again I would have liked to have said a lot more about.