Proof that time exists

Is time just an axiom? Or can it be proven to exist?

Correct me if I'm wrong, but our whole understanding of the universe is based on directly observing the world and building out axioms that are consistent with our observations.

Unfortunately, axioms cannot be proven to be true (I believe Godel's incompleteness theorem proved that). So if time is an axiom, then it is not provable.

Time also cannot be directly observed either, unlike perceiving an object (seeing a moon, feeling the pressure of the water, etc.)

Time is also irrelevant for some physical concepts such as Work.

With all of that said, my question is can time be proven to be/exist? And a secondary question of are there physical/mathematical theories being developed that take it as their axioms that time does not exist?

Answer

Check Shape Dynamics. In some sense, time doesn't exist in this formalism.