Abstract
While Alan Turing is best known for his work on computer science and cryptography, his impact on the general theory of computable functions (recursion theory) and the foundations of mathematics is of equal importance. In this article we give a brief introduction to some of the ideas and problems arising from Turing’s work in these areas, such as the analysis of the structure of Turing degrees and the development of ordinal logics.
Highlights
David Hilbert (1862-1943), one of the most prominent mathematicians of his time, developed in the 1920's an ambitious programme for laying the foundations of mathematics1
New firm foundations were much needed in the wake of the discovery of several paradoxes involving some of the most basic that attract us most when we apply ourselves to a mathematical problem is precisely that within us we always hear the call: here is the problem, search for the solution; you can find it by pure thoughts, for in mathematics there is no ignorabimus. (Hilbert, 1925)
It appears that Hilbert was convinced that his proposed system for the foundation of mathematics, or perhaps some extension of it, could be (Hilbert, 1928): Complete: Given a formula φ, either T ⟝ φ or
Summary
On Turing’s legacy in mathematical logic and the foundations of mathematics a079. David Hilbert (1862-1943), one of the most prominent mathematicians of his time, developed in the 1920's an ambitious programme for laying the foundations of mathematics. The need for formalization and strict derivation of mathematical statements according to explicitly stated logical rules was the only way, according to Hilbert, to be able to reason about ideal elements, such as actual infinite sets, while avoiding the paradoxes To attain this goal, Hilbert proposed a foundation for mathematics that he called proof theory, whose goal was to build a formal system consisting of that should be obtained by purely finitary means (i.e., not involving ideal elements), for he believed that only finitary statements are firmly grounded. Does not give any argument of why this is so He seemed to believe that, in addition to being complete and consistent, some formal system based on first-order logic and strong enough to encompass all ordinary mathematics could be Decidable: There is a definite method, or mechanical process, by which, given any formula φ, it can be determined whether or not φ is provable in the system. Let us observe that completeness implies a positive solution to the Entscheidungsproblem (assuming the axioms are given effectively)
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