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Computable Numbers: Difference between revisions
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== Behind the scenes == | == Behind the scenes == | ||
Hilbert's program was an attempt to provide a secure foundation for mathematics, that is, a secure set of symbols and rules that all statements can be defined in, a proof that all true statements can be proven in this system, the system to be consistent, and there to be a single procedure to follow for deciding the truth or falsity of any mathematical statement in general. [[Kurt Gödel]] published his first and second [[incompleteness theorems]] in his [[1931]] paper ''On Formally Undecidable Propositions in Principia Mathematica and Related Systems I'', which were largely seen as ending any hopes for this project, casting doubt on the consistency of any such system, and definitively removing its completeness. Turing, in his [[1937]] paper ''On Computable Numbers, with an Application to the Entscheidungsproblem'' removed the last remaining remnant of the project, by showing that there is no general algorithm for finding the truth or falsity of any given mathematical statement. | |||
== External links == | == External links == | ||
* '''[https://doi.org/10.1112/plms/s2-42.1.230 Full text of the paper]''' | * '''[https://doi.org/10.1112/plms/s2-42.1.230 Full text of the paper]''' | ||
[[Category:Mathematics from the real world]] | [[Category:Mathematics from the real world]] | ||
[[Category:Documents from the real world]] | [[Category:Documents from the real world]] | ||
