0

I have yet another post about what is model theory doing and why is it valid; I hope I can be coherent.

(1) https://mathoverflow.net/questions/23060/set-theory-and-model-theory

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

(3) Is an interpretation just a homomorphism between theories?

(4) https://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent

are in a similar vein and since some of them are over 10 year old I decided to make a new post for this discussion, although I want to clarify some of the points people are making there; I will quote the relevant sections.

The Problem

So thankfully it seems Im not alone in finding model theory hard to understand. The confussion for me is "where does it exist"? Is it "meta" or is it internal to ZFC. I think that maybe this is hard because it is a question that actually depends on your foundations / philosophy.

Im adopting a Curry type formalist perspective as a minimal position. My diognosis is that this question is maybe as much about languages as much as models. The question of what a model is will depend on what we think a language is. My position is that a language is the primitive concept that I assume. All else will be built on an intuitive and assumed grasp of what it means to have symbols, move symbols, differentiate symbols etc

A specific instance that im stuggling to understand is how I can interpret in this way the internalization of models of ZFC e.g. in independence proofs.

My understanding

I will outline my attempt at an answer and maybe someone can tell me how to fix it:

We have two languages, these are primitive. One of them is ZFC. A model is a translation of one language into ZFC. It is at the meta-level,it is an act of human computation, specification, at the level of the linguistic primitive of moving symbols around. It needs to satisfy some properties such as "true" in the first language $\implies$ "true" in ZFC.

Here we could take truth to be a primitive associated to languages or define it in terms of the primitives of the language (derivable in a proof system).

I will call this process intuitive modelling for later reference.

If this is what is meant then it is clear that this is not formal (it doesnt exist within a given language, unless we postulate a universal language maybe ...?) it is intuitive at the level of our primitive ideas of language.

Does anyone mean this by model or am I missing the point?

If this is not what they mean then I can only see that they mean something much more mundane, a language is litterally a type of set in ZFC. A model is litterally a function within ZFC.

The only problem with this is then how do we view statements of independence. What is a "model of ZFC + CH"? It cannot be a set in ZFC I beleive for technical reasons (Godel?); but it cannot be a set in ZFC for logio-philosophical reasons, its clearly circular ZFC is a set in ZFC?

Responses

The first response I see is to sort of take mathematics as given and then just say model theory is a part of it and dont worry

A theory is a set of sentences; a model is a "piece of the mathematical world" that satisfies the sentences (axioms) of the theory. ... What are number? What is $\mathbb{N}$ ? It is the set of natural numbers; how we "known" it? How we describe it? With usual mathematical jargon. (2)

or in the first link

if you ask "why are model theorists justified in using sets?" then I ask back "why are number theorists justified in using numbers?" (1)

This is valid but not really helpful for me I guess.

Stefan Hoffelners response in (1) is excellent but im still troubled by these "codings"

As you know, one can code the symbols of first order logic within set theory and, as a consequence, the whole model theory can be carried out in ZFC.

Is this coding not an example of intuitive modelling as I have described above?

Thanks

Im clearly very confused about this, I have tried to be succinct and coherent, questions and clarifications are welcome. Thanks for any engagement.

Riley Moriss
  • 19
  • You might find this question on MathOverflow interesting. – Joe Dec 30 '23 at 16:17
  • Indeed that is very interesting, thank you. I think Im realising that the real problem is this, with model theory you have a meta-theory and a subject theory. People model ZFC in ZFC (do they, or ZFC+I?) is making the meta and subject theory the same. Which is not prima facie valid. – Riley Moriss Dec 31 '23 at 15:49
  • Why do you think it is not valid? If you want to take a Platonistic approach, then the $\mathsf{ZFC}$ axioms describe the "true universe of sets" $(V,\in)$, and thus a set model $(M,\varepsilon)$ of $\mathsf{ZFC}$ (where $M$ belongs to the true universe and $\varepsilon$ is a relation on $M$) is an interpretation of $\mathsf{ZFC}$ (in the technical sense of logic), but is not the intended one. The intended interpretation is $(V,\in)$, which Platonists believe they have an understanding of, even though no reasonable axiom system can "pin it down". – Joe Dec 31 '23 at 15:55
  • I think the Platonistic approach helps to motivate the situation, even if you ultimately reject it as philsophically unsatisfactory. If you want to describe the formalist counterpart to the Platonistic interpretation, please let me know. – Joe Dec 31 '23 at 15:56
  • This platonic explination you have given makes a lot of sense at an intuitive level. Im interested in your formalist counterpart, becuase thats what makes me feel "sure". The answer given in this post https://mathoverflow.net/a/249006/519939

    Describes how Im thinking things work. [Cont]

     – Riley Moriss Dec 31 '23 at 16:47
  • The point is that there are two ZFC's, meta and subject, and we are not justified in conflating the two (im not convinced that we are at least); hence statements like ZFC doesnt have a model are proved in the meta about the subject and so are dishonest; the correct statement is that the subject-ZFC doesnt have a model in the meta-ZFC..? – Riley Moriss Dec 31 '23 at 16:47
  • The terms "metatheory" and "object theory" don't have formal definitions in mathematics: they are just labels to help keep our ideas straight. If you believe it is acceptable to reason in $\mathsf{ZFC}$ (or at least, use reasoning that can be formalised in $\mathsf{ZFC}$), then I don't think there is anything wrong with using $\mathsf{ZFC}$ to reason about $\mathsf{ZFC}$. It's not that there are two $\mathsf{ZFC}$'s – rather, it is that $\mathsf{ZFC}$ is expressive enough to reason about itself. – Joe Dec 31 '23 at 17:00
  • Compare say with the group theory axioms. We can reason about the group theory axioms, and models of group theory axioms (i.e. groups), but this reasoning can't be formalised in group theory itself – usually it would be formalised in $\mathsf{ZFC}$, or some other foundational theory. However, we can talk about models of $\mathsf{ZFC}$ in $\mathsf{ZFC}$ (even though, by Gödel, if $\mathsf{ZFC}$ is consistent then it can't prove that it has a model). – Joe Dec 31 '23 at 17:01
  • For instance, $\mathsf{ZFC}$ proves statements like "if $\mathsf{ZFC}$ has a model, then it has a countable model" (or rather, $\mathsf{ZFC}$ proves the "translation" of that statement into the language of set theory). – Joe Dec 31 '23 at 17:01
  • Okay, so there is a sense in which there are "two" $\mathsf{ZFC}$'s. The first is the "naive" version, where we use English words like "set" and "true" and so on, without much philosophical care, and we take quite a casual attitude to its syntax, for instance by freely expanding it to include the symbol $\subseteq$. The proofs in this naïve version are written in ordinary English. However, logicians are convinced that all of these ordinary proofs could, in principle, be written as formalised proofs. – Joe Dec 31 '23 at 17:06
  • These formalised proofs would be written in a strict logical syntax, and wouldn't even use the word "set" at all. Logicians are convinced that you don't need to have a Platonistic committment to the existence of sets in order to make sense of these everyday proofs, precisely because those proofs can be thought of as mere shorthand for the long (perhaps meaningless) formal proofs. – Joe Dec 31 '23 at 17:07
  • So even if you think there are two $\mathsf{ZFC}$'s, an everday one and formal one, they are still inextricably linked. However, I believe that most of the time logicians say $\mathsf{ZFC}$, they mean the formal version. – Joe Dec 31 '23 at 17:11
  • So this post https://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent is extremely relevant to what Im thinking of. The sense in which there are two ZFC's, if model theory is formal, is worse than what you say; ZFC(meta) is not a set, it needs to be internalised. This is the reason it is not prima facie possible to use ZFC to reason about ZFC. ZFC is about sets, ZFC is not a set. – Riley Moriss Dec 31 '23 at 17:18
  • This is the reason for my subtitle: Which set is ZFC. When you internalise ZFC you have to pick a set. Then you prove things about this set. Not the language itself. The two are conflated. I beleive through reading this site I am correct about this (see the above link). They might be conflated for good reason, and there might be ways of avoiding this conflation in cases, but none the less. – Riley Moriss Dec 31 '23 at 17:21
  • Sure, $\mathsf{ZFC}$ is not a set, and the language of $\mathsf{ZFC}$ doesn't contain a "noun" for $\mathsf{ZFC}$. But we can still use $\mathsf{ZFC}$ to reason about $\mathsf{ZFC}$, in an indirect way. To say that, say, $\mathsf{ZFC}$ is consistent, can actually be thought of as a statement in the language of arithmetic (a result due to Gödel), and $\mathsf{ZFC}$ can reason about the natural numbers ${0,1,2,\dots}$ (since those natural numbers can implemented as sets). – Joe Dec 31 '23 at 17:25
  • Moreover, we are confident that every statement that $\mathsf{ZFC}$ proves about its internal version of the natural numbers is true of the actual natural numbers, i.e. we are confident that $\mathsf{ZFC}$ is arithmetically sound. – Joe Dec 31 '23 at 17:25
  • I think what you're calling "intuitive modeling" is what is usually called "formalization in set theory" or "interpretation in set theory" and it has little to do with models, much less with model theory, other than that model theory can be interpreted in set theory (just like algebraic geometry, complex analysis, elementary number theory, category theory, point set topology or any other branch of math can be interpreted in set theory). – spaceisdarkgreen Dec 31 '23 at 22:00

1 Answers1

1

Model theory (of first order theories) just looks at sets with some relations and functions on them that are appropriate for interpreting formal sentences elaborated on previously specified signatures (some symbols for relations and functions) and according to some (common) grammatical rules (those of first order theories)

Familiar examples include the theory of groups, whose signature includes a binary function symbol (to be interpreted as the product), a unary function symbol (to be interpreted as the inverse function), and a nullary/zero-ary function symbol (to be interpreted as the group identity), and the theory of rings, whose signature you should be able to guess. Now, any set whatsoever 'equipped' with appropriate functions and symbols can in principle be seen as a structure to interpret the sentences formulated in these languages, but only those satisfying - and this is the fundamental concept/notion upon the whole subject is built - some given sentences (the axioms of the theory) are of interest

What about set theory, then? They (there are many) (generally?) are first order theories themselves, so of course they are susceptible to model-theoretic analysis/treatments: their signature consisting of a binary relation symbol, to be interpreted as membership, and their axioms a bunch of sentences elaborated on the usual way of first order theories, a model of a set theory is simply going to be a set with a binary relation

Perhaps we need to take a step back to stress an important point: on the one hand we have symbols, signatures, languages, sentences, axioms, etc., and on the other we have actual objects (sets), with actual functions and relations on them. This may cause some confusion at first: isn't everything on (the usual) set theories just sets? The strict answer is 'yes', so the difference in hands is less of a literal one and more of a 'moral' one: we somehow ought to think of these things as different, for they are qualitatively so, or at least because or intentions towards them are different

Still, what about set theory being used to formulate model theory to begin with? Isn't that somehow "circular"? Well, we do use natural languages to do/study linguistics, and we do cosmology inside the (our?) universe: it's kind of just a fact of life we have to learn to deal with... ok, this sounds sort of dismissive of the problem, so let's think a bit more

Incompleteness issues prevent us from actually proving, from a given set theory, that this set theory has a model, but we can simply assume "Let there be lig... a model", and take it from there: from this moment on, we have a structure, a bona fide object to work with inside our set theory. Notice though that it being a model only ensures it satisfies the axioms insofar as 'satisfy' speaks of its elements behaving in such and such a manner amongst themselves and the specified (functions and) relation(s), it doesn't necessarily mean it somehow relates to the other objects of the ambient/underlying set theory in any 'meaningful way': the most commonly cited example is the powerset, in that an element of the model is going to have a powerset 'inside the model', and this doesn't necessarily have to be equal to the powerset as it exists 'outside'

There are many other intricacies and technicalities regarding model theory of set theories (the huge subject of set theory as commonly now understood basically is model theory of it), but I hope you can find some comfort with the main take away that set theories are first order theories like any other (well, not quite, but anyway), and so it's perfectly legitimate to do model theory of them

ac15
  • 984
  • Thanks for the response. I would like to ask if my distinction between models inside ZFC (As you seem to be referring to?) And intuitive modelling makes sense? It seems like you are treating first order theories as objects inside your set theory. In which case there should be a distinction between two ZFCs, the one that we use and the one that is "inside" that one. Am I understanding? Then you just identify them because they are very similar? – Riley Moriss Dec 29 '23 at 07:13
  • 1
    To be honest, I gave the answer based solely on trying to answer the bit "what is model theory doing and why is it valid[?]": I didn't go check the previous questions, and the rest of the text in the present question made essentially no sense to me (as is the case with this comment-question now) – ac15 Dec 29 '23 at 16:29
  • My claim is that model theory is inherently informal, especially when modelling ZFC. A first order language is NOT an object of set theory, it is primitive, when you make it one by labelling a set with the elements of the theory you have done something that is not formal. If you say that a theory is a set then you have to explain how the set theory contains itself which is circular or relies on the translation that I just mentioned. Hope that helps? – Riley Moriss Dec 30 '23 at 01:51
  • 1
    Model theory is not inherently any less formal than any other bit of/subject in mathematics. Specifically, there's no point in thinking this would apply 'especially when modelling ZF(C)': again, set theories are a priori first order theories like any other, nothing more, nothing less. The sentences of a first order language are just strings, and arithmetic theories and set theories have reasonable ways to encode/internalize/'indirectly speak of' such things: think of the way high level programming languages are 'translated' into machine language when running on a computer (imperfect [cont.] – ac15 Dec 30 '23 at 01:58
  • analogy, I know). This process of arithmetization of syntax is not 'not formal' (it might, in fact, be the single most formal thing that anyone ever does in non-purely-mechanized mathematics, but that's a bit beside the point). There's really nothing circular/wrong about a theory being used to encode/arithmetize itself: it's just manipulating bits of text, 'meaningless ones', as it stands. I mean, there are compilers that can compile themselves, so none of this should come as a surprise. Also very important: there's doesn't seem to be any sense in saying that a language "is primitive" [cont.] – ac15 Dec 30 '23 at 02:06
  • maybe you're confusing the fact that each language has some symbols to which we refer to as 'primitives', but really I'm not sure – ac15 Dec 30 '23 at 02:09
  • So I think the crux of the thing is here

    "This process of arithmetization of syntax is not 'not formal' "

    When I say formal I mean happening inside a first order language. What do you mean? I can't see how you can consider this translation as happening anywhere but externally on a human level. By primitive I just mean logically prior (axiomatic or first or something)

     – Riley Moriss Dec 30 '23 at 06:11