I do not know who is the author of this bad joke, but it is appalling that even qualified mathematicians often say that Cantor’s proof of existence of transcendental numbers is non-constructive, as opposed to Liouville’s proof.

Let me recall that Liouville proved that the infinite sum is transcendental, while Cantor proved that the set of algebraic numbers is countable, but the set of reals is uncountable.

Dear colleagues, Cantor’s proof is less “constructive” than Liuoville’s only on emotional or aesthetic level. Mathematically, **both are equally constructive**.

Indeed, the only way I can imagine of defining a real number “constructively” is providing an algorithm for computing rational approximations to it of any prescribed precision. More formally, call a real number *constructible* if there exists an algorithm (a Turing machine) which, having a natural number at the input, produces at the output a rational approximation to with precision . I do not know another reasonable (non-equivalent) definition of a constructible real number; if anybody knows, I would be happy if he shares it with me.

Of course, the Liouville number is constructible, the good rational approximations being the partial sums of the infinite series.

Further, call a sequence constructible if there exists an algorithm, which, having natural and at the input, produces a rational approximation to with precision at the output.

Now two simple exercises.

- Show that there exists a constructible sequence containing all algebraic numbers.
- Show that for any constructible sequence, there exists a constructible number not contained in this sequence.

A careful examination of Cantor’s proof reveals that its first part solves the first exercise, and the second part solves the second exercise.

I would never waste time for recalling these trivialities in my precious blog if this misunderstanding were not so widespread even among professional mathematicians.

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This entry was posted on June 14, 2010 at 3:33 am and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
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June 14, 2010 at 6:03 am |

waist => waste.

June 14, 2010 at 10:34 am |

Спасибо, поправил.

June 14, 2010 at 5:39 pm |

Kantor -> Cantor

(But then, he started his life in Cyrillic.)

June 14, 2010 at 8:35 pm |

Merci, c’est corrigé.

December 8, 2010 at 6:39 am |

Do you really need rational approximations to imagine real algebraic numbers?

March 6, 2011 at 6:14 pm |

I need rational approximations to imagine

realnumbers.