[necovek]

What frequently happens when we recombine axioms like that is that they end up leading to inconsistencies or contradictions.

Do you know if this topos with every total function on real numbers is continuous has been constructed and proven to be a viable set of axioms? If so, I am curious about the source.

My go to example still remains the one of hyperbolic geometry and axiom of parallel lines, so the more approachable examples I can get, the better.

[black_knight]

Sure. These toposes are well known, and proven to be consistent (relative to set theory). For instance Hyland’s effective topos, or Johnstone’s topological topos. The ideas are that these toposes either require everything to be computable, or continuous in some greater sense.

There is also this blogpost by Amdrej Bauer, which can be seems as exploring how it is to be such such a topos: https://math.andrej.com/2006/03/27/sometimes-all-functions-a...