Continuing with diagonalization

Diagonalization uses some interesting logic to state that if you postulate a statement, an contradictory statement also exists, and you cannot have an algorithm that can solve both contradictory states. But there is a catch. What if there are no contradictions in nature. What if the nature is always right, and found its solutions. Can natural numbers display alternate properties. Can we have unconstrained input set for Turing Machines. If a given state is achieved, what if an alternate state does not exists. In that case Hilbert's second principle can still be realized, and you can create a set of axioms that prove all mathematical statements.