I'm studying Skolem's Paradox. Cantor's Theorem: for any (arbitrary) set S, no mapping surjects S onto its power set P(S). Consequently, there exist uncountable sets. Skolem's Theorem: if some collection of first-order sentences has any infinite model, then it has a countable model. The paradox: how can (e.g.) ZFC, a first-order theory, have a countable model and imply that there uncountable sets? I understand Cantor's Theorem, but not so much Skolem's Theorem.
Timothy Bays spells it out (my paraphrase):
Let T be ZFC. Let M be a model of ZFC. Since T ⊢ ∃x x is uncountable, there must be some m ∈ M such that M ⊨ m is uncountable. But, as M itself is only countable, there are only countably many m ∈ M such that M ⊨ m ∈ m. Thus there seems to be a contradiction.
(1) what is "m^" and what does it mean to say "m ∈ M"? -- I ask this because I understand what x is such that x ∈ T: it is a first-order sentence (e.g. one of the axioms in ZFC), but I guess I don't understand models. What does it mean to say "a model with a countable domain?"
(2) Can someone explain in plain English what the relevant Skolem (-Lowenheim) Theorem is saying?
(3) General comments?
Thanks! Feel free to be thorough (though clear), I'd appreciate it.