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
(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.