Logic | Mathematical Logic
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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.
In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true in every possible interpretation . Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies , and continues to be used in that alternate sense. A formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions .
Tautology (logic)
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 . According to naive set theory, any definable collection is a set . Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves .
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects , facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory , which is semantic in nature.
Logic | Set Theory