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Letter Games: a metamathematical taster

Published online by Cambridge University Press:  17 October 2016

A. C. Paseau*
Affiliation:
Wadham College, Oxford e-mail: alexander.paseau@philosophy.ox.ac.uk

Extract

Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

Type
Articles
Copyright
Copyright © Mathematical Association 2016 

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References

1. Gödel, K., ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’, Monatshefte für Mathematik und Physik, 38, (1931) pp. 173198, translated by van Heijenoort, J. as ‘On formally undecidable propositions of Principia Mathematica and related systems I’ and reprinted in Collected Works Vol. 1: Publications 1929-1936, Feferman, S. et al. eds, Oxford University Press (1986) pp. 144-195.Google Scholar