Logic | Mathematical Logic
Gödel's incompleteness theorems. Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.
The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic).
For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. Tautology (logic) Is sometimes used to denote an arbitrary tautology, with the dual symbol (falsum) representing an arbitrary contradiction.
Tautologies are a key concept in propositional logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (or, equivalently, whether its negation is unsatisfiable). Russell's paradox. In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction.
The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen. In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF). Informal presentation Proof theory. Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.
History Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics. Logic | Set Theory.