A question about FOL theories and models

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.
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What we call 'I' is just a swinging door which moves
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Shunryu Suzuki
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