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hairyfigment comments on Second order logic, in first order set-theory: what gives? - Less Wrong Discussion
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Comments (19)
Set theory has countable models? Countable according to who, or according to what system?
I've grappled with model theory in the past, but only long enough to convince myself that my interpretation of Godel and Lob held water. Poizat seems to say that set theory need not have a countable model unless the language has (at most) countably many symbols; now in practice this is always true, but for set theory we keep pretending that it isn't.
ZFC amounts to a binary relation "is an element of", satisfying some axioms. A countable model of ZFC is a binary relation on the integers 1,2,3,... satisfying the axioms. According to set theory such a relation exists, for instance this is a consequence of the Lowenheim-Skolem theorem. This relation is not computable.
According to ZFC + "ZFC has models."