Hilbert’s Attempt at Recovery

Abstract

Hilbert’s Program was a promising formalistic attempt to recover mathematics. It would use formal axiomatic systems to put mathematics on a sound footing and eliminate all the paradoxes. Unfortunately, the program was severely shaken by Gӧdel’s astonishing and far-reaching discoveries about the general properties of formal axiomatic systems and their theories. Thus Hilbert’s attempt fell short of formalists’ expectations. Nevertheless, although shattered, the program left open an important question about the existence of a certain algorithm—a question that was to lead to the birth of Computability Theory.

If something is consistent, no part of it contradicts or conflicts with any other part. If something is complete, it contains all the parts that it should contain. If something is decidable, we can establish the fact of the matter after considering the facts.