[FOM] Who was the first to accept undefinable individuals in mathematics?
William Tait
williamtait at mac.com
Wed Mar 11 11:50:57 EDT 2009
.
On Mar 10, 2009, at 7:38 AM, W. Mueckenheim wrote:
> Until the end of the nineteenth century mathematicans dealt with
> definable numbers only. This was the most natural thing in the world.
The assumption by Bolzano (1817) and Cauchy (1821) in proving the
intermediate value theorem that every Cauchy sequence of rationals
determines a real certainly does not display a concern about
definability. But, in any case, definable in what language?
> An example can be found in a letter from Cantor to Hilbert, dated
> August 6, 1906: "Infinite definitions (that do not happen in finite
> time) are non-things. If Koenigs theorem was correct, according to
> which all finitely definable numbers form a set of cardinality
> aleph_0, this would imply that the whole continuum was countable, and
> that is certainly false." Today we know that Cantor was wrong and
> that an uncountable continuum implies the existence of undefinable
> numbers.
Surely Cantor was wrong only in the sense that he didn't point out
that the notion of definability cannot be absolute, but depends upon
the language.
Bill Tait
More information about the FOM
mailing list