Is it possible to develop a set of axioms for solving any problem, that are certain to work? Similar to problem solving strategies or proof strategies, though a set of steps that work indefinitely if applied effectively?
And I mean elementary proofs or problems that have already been solved before.
A simple-seeming class of problems - Diophantine equations - cannot be solved by an algorithm of any sort.
A Diophantine equation is an equation of the form $P(x_1,x_2,\dots,x_n)=0$, where $P$ is a polynomial with integer coefficients and you are seeking integer solutions. There is no algorithm to determine if such an equation has a solution.
Turns out, you can restrict the question to polynomials of degree $4$ at most, and it is still unsolvable.
This is called Hilbert's Tenth Problem.