alright just for fun: AMA but only about math, will attempt to speed-explain stuff with as few symbols and equations as possible and see what happens
(esp happy to field questions about stuff that seems basic to you and that you feel like you should've gotten a long time ago!)
Conversation
what's the whole deal with "models"?
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hmm can you elaborate? like models of physical situations, that kind of thing? like what's the deal about the relationship between "models" and "reality"?
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uh i think this kind of thing? although i know very little about it?
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oh model theory yeah sure. so, there's a general pattern in modern mathematics where you define some class of objects in terms of them having some operations satisfying some axioms, e.g. groups, rings, fields, etc.
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model theory is studying this defining procedure itself: there are these things called "first-order theories" (and "the first-order theory of groups" etc. is an example) and "models" of these are sets having some operations satisfying some axioms specified by the theory
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it turns out there are a lot of really wacky things you can say about how models of first-order theories behave in general and model theorists study those things
one major application is that there is a "first-order theory of arithmetic" and a "first-order theory of sets"
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so model theory implies various things about arithmetic and set theory, for example that there are "nonstandard" models of arithmetic, and also that there is a *countable* model of ZF set theory. really wacky stuff
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in the context of examples like arithmetic and set theory there's a real need to cleanly separate "syntax" from "semantics" - syntax is like the word games of how we write down proofs, and semantics is what those proofs *mean*. models provide semantics
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so it can be quite surprising to learn that there are different inequivalent semantics for (first-order) arithmetic and set theory but this turns out to be a general feature of many first-order theories