In many systems of logic, denotations are truth values. Computational logic Ī semantics is a system for mapping expressions of a formal language to their denotations. Today recursion theory is mostly concerned with the more refined problem of complexity classes-when is a problem efficiently solvable?-and the classification of degrees of unsolvability. Recursion theory captures the idea of computation in logical and arithmetic terms its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing, and his presentation of the Church–Turing thesis. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem, through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis, to the modern debate on large cardinal axioms. The various attempts to carry this out met with failure, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems. Mathematical theories were supposed to be logical tautologies, and the programme was to show this by means of a reduction of mathematics to logic. The latter was a major concern in early 20th century mathematical logic, which pursued the program of logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic.
Major subareas include model theory, proof theory, set theory, and computability theory. Mathematical logic is the study of logic within mathematics.
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