>>11640213>Is the entire building of math based, finally on faith their axioms are self evident but can't be proven in the end?There are systems of arithmetic which are provably consistent, but they are weaker than even the Peano axioms, if you want to know.
A good way to think of what axioms are simply to consider them definitions. Euclids(well, really hilberts axioms) define what a flat plane is and give you grounds to start solving problems and proving theorems.
But regardless, the set theoretic and logical problems are irrelevant to a significant amount of mathematics. If we go through another foundational crisis and ZFC turns out to be self contradictory, another system will be established and almost everyone who worked in geometry or in analysis will not even notice the difference. All the facts of life you know and love such as the classification of 2d closed manifolds or the prime number theorem, Gauss Bonnet theorem, Riemann-Roch with all its generalizations, Central Limit theorem, Stokes Theorem, Fermats Last Theorem, etc will likely be unaffected.
