The List of Hilbert’s Twenty-Three Problems: Hilbert’s Second Problem

The Compatibility of the Standard Axioms of Arithmetic: Prove that the axioms of arithmetic are consistent.

Hilbert’s second problem was to prove that arithmetic is consistent, that is, that no contradictions arise from the basic assumptions he had put forth in one of his papers. 

This problem has been partially resolved in the negative: Kurt Gödel showed with his incompleteness theorems in 1931 that it is impossible to prove the consistency of a system called Peano arithmetic using only the axioms of Peano arithmetic. Mathematicians debate whether Gödel’s work is a satisfying resolution to the problem.

Πηγή: abakcus