Re: A question about FOL theories and models


Nam Nguyen wrote:
ZF(C), after all, is just one theory out of infinite number
of 1st order ones. (Formally, FOL framework never insists that
we have to know about ZFC to formalize a different theory.)

Assuming we've formalized a theory G of "geometry", how could we prove
that the 5th "postulate" - as an axiom - is unprovable in G, without
mentioning anything about ZF(C)? In other words, how could we possibly
come up with a specific model of G in which the 5th is false? Thanks.
--

If you're going to do model theory, you have to decide what metatheory
you're going to work in. You can prove that there exists a model of
Hilbert's axioms with the parallel postulate negated in fourth-order
Peano arithmetic, and any stronger theory, such as ZFC.

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What we call 'I' is just a swinging door which moves
when we inhale and exhale.
Shunryu Suzuki
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