[FOM] Cantor on Richard's Paradox

William Tait williamtait at mac.com
Tue Jul 3 09:29:45 EDT 2007

On Jul 2, 2007, at 10:20 AM, laureano luna wrote:

> My questions are:
> Cantor seemingly believed the set of possible
> definitions of reals was not countable: how was this
> possible?

Cantor wasn't speaking of definability in some particular language. I  
expect that the following is behind his statement:

Let A be the set of reals definable in the language L. Apply (a  
modification of) Cantor's nested intervals argument to define a real  
not in A. This definition is in a language L' including L (in which  
one can speak of definability in L). In this way, a sequence  <L_ 
{alpha> : alpha < omega_1> of 'languages' is defined such that the  
corresponding sequence of sets A_{alpha} is strictly increasing. The  
'paradox' arose from not realizing that L' is not L.

Bill Tait

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