FOM: Cantor's theorem of little interest in constructive math

Miguel A. Lerma mlerma at
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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