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Jack Campin - bogus address wrote:
> > 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.
this is one place where matroid theory
has lain a very strong foundation
pregeometric models can be constructed
purely in terms of sheafifications
and we can extend the four axioms
in many independent ways
similar questions were the motivation
for rota's initial introduction of the machinery
this has been categorified
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galathaea: prankster, fablist, magician, liar
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