[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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