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A survey of mathematical logic, part II: post-1931

Published online by Cambridge University Press:  01 August 2016

G. T. Q. Hoare
G. T. Q. Hoare*
Affiliation:
Dr Challoner’s Grammar School, Amersham, Bucks HP6 5HA
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Extract

In part I of this article [1], we described developments in mathematical logic before 1931. Gödel’s famous paper On formally undecidable propositions of Principia Mathematica and related systems, was published in January of that year. In it Gödel established two theorems:

  1. (1) Incompleteness: For any consistent theory at least as strong as arithmetic there are sentences which can neither be proved nor disproved in the theory.

  2. (2) Unprovability of consistency: The consistency of such a theory cannot be proved within the theory.

We are going to outline, with minor adjustments to notation, Gödel’s actual proof of (1) given, in translation, in [2, pp. 5–38].

Type
Articles
Information
The Mathematical Gazette , Volume 80 , Issue 488 , July 1996 , pp. 286 - 297
Copyright
Copyright © The Mathematical Association 1996

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References

1. Hoare, G.T.Q., A survey of mathematical logic, part I: pre-1931, Math. Gaz. 80 (March 1996) pp. 83–91.CrossRefGoogle Scholar
2. Davis, Martin (editor), The undecidable, Raven Press (1965).Google Scholar
3. Herken, Rolf (editor), The universal Turing machine, a half-century survey, OUP (1988).Google Scholar