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Chris Ormell's article

Published online by Cambridge University Press:  23 January 2015

Bob Burn
Bob Burn*
Affiliation:
Sunnyside, Barrack Road, Exeter EX2 6AB, e-mail:[email protected]
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Extract

In [1], Chris Ormell raised a question about the uncountability of the real numbers. Ormell affirmed that only a countable collection of numbers may be defined by a finite number of words, a notion that stems from a letter of Richard (1905), reproduced in [2, p. 142]. This appears to conflict with Cantor's proof (1874) of the uncountability of the reals [3, p. 839]. Poincaré, whom Ormell claims as a witness in his defence, said ‘Now as is well known, Cantor proved that the continuum is not denumerable; this contradicts the proof of Richard. The question therefore arises which of the two proofs is correct. I maintain that they are both correct and that the contradiction is only apparent.’ [3, p. 1072] Poincaré then supplied a new proof that the points on an interval are not countable.

Type
Matter for Debate
Information
The Mathematical Gazette , Volume 94 , Issue 529 , March 2010 , pp. 112 - 113
Copyright
Copyright © The Mathematical Association 2010

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References

1. Ormell, Christopher, Can we understand uncountability? Math. Gaz., 92, (July 2008) pp. 252–256.Google Scholar
2. van Heijenoort, J., From Frege to Godel, Harvard (1967).Google Scholar
3. Ewald, W., From Kant to Hilbert, Vol. II. Oxford (1996).Google Scholar
4. Smith, H. J. S., On the integration of discontinuous functions, London Maths. Soc. Proc., 6, pp. 140–153 (1875).Google Scholar