I'm reading Beginning Mathematical Logic: A Study Guide by Peter Smith (can be downloaded here). I came across the following sentence (pg. 17) and can't understand what Smith means:
(b) Turning to semantics: the first key idea we need is that of a model structure, a (non-empty) domain of objects equipped with some properties, relations and/or functions. And here we treat properties etc. extensionally. In other words, we can think of a property as a set of objects from the domain, a binary relation as a set of pairs from the domain, and so on. (Compare our remarks on naive set theory in §2.1.)
From my limited (informal) knowledge of model theory I've observed from elsewhere, what I envision is that you could take, for example, real numbers and map them to Dedekind cuts and take addition and multiplication and map them to functions on $\mathbb{R}$ (extensionally).
The first thing I'm noting about this:
- We use ZFC to describe the semantics of something like the real numbers (again, this is my guess from what I've seen about model theory elsewhere)
- How can we make a model of ZFC? Surely it can't be extensional (such as when talking about $\mathbb{R}$) - there is no "set of all functions" in ZFC. Besides, wouldn't every object (set) in the domain of discourse just get interpreted as itself?
Is model theory ever used when talking about models of ZFC? What would it even mean to have a model of ZFC itself?
Just throwing this out here (I'm obviously very naive). Could someone shed some light?