Editing David Hilbert (section)

====Gödel's work====
Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

[[Kurt Gödel|Gödel]] demonstrated that any consistent formal system that is sufficiently powerful to express basic arithmetic cannot prove its own completeness using only its own axioms and rules of inference. In 1931, his [[Gödel's incompleteness theorem|incompleteness theorem]] showed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely [[finitary]].

Nevertheless, the subsequent achievements of proof theory at the very least ''clarified'' consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of [[recursion theory]] and then [[mathematical logic]] as an autonomous discipline in the 1930s. The basis for later [[theoretical computer science]], in the work of [[Alonzo Church]] and [[Alan Turing]], also grew directly out of this "debate".<ref>{{Cite journal |last=Reichenberger |first=Andrea |date=31 January 2019 |title=From Solvability to Formal Decidability: Revisiting Hilbert's "Non-Ignorabimus" |url=https://scholarship.claremont.edu/jhm/vol9/iss1/5 |journal=Journal of Humanistic Mathematics |volume=9 |issue=1 |pages=49–80 |doi=10.5642/jhummath.201901.05 |s2cid=127398451 |issn=2159-8118|doi-access=free }}</ref>