FOM: Cantor's theorem of little interest in constructive math
Miguel A. Lerma
mlerma at math.northwestern.edu
Wed Feb 14 16:49:27 EST 2001
> Here's a familiar, positive, typical example: Classically, one might
> prove the existence of a trascendental by noting that the reals are
> uncountable and the algebraic reals are countable. We don't use the
> cardinalities in a constructive proof because it is easy to construct a
> transcendental number directly.
The existence argument based on cardinalities may be closer
to a constructive proof than usually assumed, as shown in
Robert Gray: "George Cantor and Transcendental Numbers",
Amer. Math. Month. 101 (1994), 819-832. The idea is that
the same diagonal argument used in the proof that the reals
are uncountable can be used to generate a specific transcendental
- we also need an effective enumeration of polynomials and
arbitrarily accurate approximations to their roots.
The same thing can be done with many "metric" arguments,
as shown by M.H. Lebesgue: "Sur certains demonstrations
d'existence", Bull. Soc. Math. France 45 (1917), 132-144).
An example of application of the idea is in M.F. Kulikova:
"A construction problem concerned with the distribution of
the fractional parts of an exponential function", Soviet Math.
Dokl. 3 (1962), 422-424.
Miguel A. Lerma
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