Gödels incompleteness theorems
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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