Sewing-Machine comments on Second order logic, in first order set-theory: what gives? - Less Wrong Discussion
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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.