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Gödels incompleteness theorems

Gödel's incompleteness theorems Two fundamental theorems in mathematical logic, proved by Kurt Gödel in 1931. The first concerns the formalization of basic arithmetic. Gödel showed that, in any logical system powerful enough to express arithmetical operations, there must exist sentences that are neither provable nor refutable in the logical system. In consequence there exist statements about arithmetic that, while true, cannot be proved in the logical system. The second theorem states that no logical system can be powerful enough to provide a proof of its own consistency. These discoveries marked a turning-point in our understanding of formal reasoning. For example, they forced the abandonment of “Hilbert's program”, i.e. of the search for a provably consistent and complete formal basis for the whole of mathematics itself.

Equally significant are the proof methods that Gödel used. One device was to encode logical formulas as numbers (see Gödel numbering), so that manipulations of formulas could be “programmed” as numerical computations. (This early exercise in numerical data representation and programming also marked the beginning of recursive function theory.) The other device was to use this numerical encoding to produce a formula that in effect asserts its own unprovability. The idea is seen in statements such as “this sentence is false”, the Cretan liar paradox, and Russell's paradox. Gödel's construction however is formal: the reference to “this sentence” is handled by the numerical encoding, without any need for vague English words. A relationship exists with the paradoxical combinator, while a similar approach can be used to show the undecidability of the halting problem as well as many results in complexity theory, where Gödel numbering is applied to Turing machines rather than logical formulas.

Gödel is also responsible for a completeness theorem.

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