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

William Tait williamtait at mac.com
Fri Mar 13 13:35:12 EDT 2009

My reference to Bolzano and Cauchy in the message below (from March  
11) isn't really sound (although the main point, that definability is  
relative to a language, is). The Cauchy sequence defining a zero of  
the continuous real-valued function f defined on [a,b] with  
f(a)<0<f(b) is  defined (by a nested interval construction) in terms  
of f. So in that sense, the zero is defined in terms of  f.

A better example is from Cantor's original paper on the non- 
denumerability of the reals. He concluded that there is a  
transcendental number between a and b (when a<b) and made a point of  
the fact that this result is significant even if it does not yield a  
definition of such a transcendental. The transcendental can in fact be  
defined in terms of an enumeration of all the algebraic numbers  
between a and b; but in Cantor's time, before Sturm's Theorem, no such  
enumeration could be defined.

Bill Tait

On Mar 11, 2009, at 10:50 AM, William Tait wrote:

> .
> 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

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