[FOM] Who was the first to accept undefinable individuals in mathematics?

[William Tait]

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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