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Monday, April 26, 2021

Kurt Gödel and the philosophy of mathematics

Kurt Gödel (1906-1978) was an Austrian-German logician best known for his incompleteness theorems in the philosophy of mathematics. Physicist Freeman Dyson said,


"Fifty years ago Gödel... proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics." (Infinite in All Directions, 1988)


The rest of this post is a quote from Gödel.


Incompleteness theorem


"The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that theses axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms." (