Editing Hilbert's program (section)

{{More footnotes|date=April 2023}}
{{Short description|Attempt to formalize all of mathematics, based on a finite set of axioms}}
In [[mathematics]], '''Hilbert's program''', formulated by [[Germans|German]] mathematician [[David Hilbert]] in the early 1920s,<ref>{{Citation |last=Zach |first=Richard |title=Hilbert’s Program |date=2023 |url=https://plato.stanford.edu/archives/spr2023/entries/hilbert-program/ |work=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |access-date=2023-07-05 |edition=Spring 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref> was a proposed solution to the [[foundational crisis of mathematics]], when early attempts to clarify the [[foundations of mathematics]] were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, [[Completeness (logic)|complete]] set of [[axiom]]s, and provide a proof that these axioms were [[consistent]]. Hilbert proposed that the consistency of more complicated systems, such as [[real analysis]], could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic [[arithmetic]].

[[Gödel's incompleteness theorems]], published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a [[computable]] set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore could not prove everything else.