Re: A question about FOL theories and models

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?

You just need to be able to construct a hyperbolic model. I haven't
seen that done without using coordinate methods (i.e. simple real
analysis) but you don't need choice or replacement, just a theory of
manifolds (and probably a constructive one, I doubt you need the
intermediate value theorem). H.G. Forder's "Geometry" describes
the construction in a pretty straightforward way.

I think topos theory can be used to address this, you might look at
Lawvere's work. But you will need *some* sort of model to prove
independence, this machinery will just let you find one that uses
the least background mathematics.

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