[FOM] Cantor on Richard's Paradox
laureano luna
laureanoluna at yahoo.es
Wed Jul 4 09:54:37 EDT 2007
Bill Tait wrote:
>
> 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.
Yes, I believe this is so too. But even so it is
strange that Cantor could think that such a hierarchy
can take us beyond the countable; we now know it seems
to find a natural stopping point at the Church-Kleene
first non constuctive ordinal, which is countable.
It looks like Cantor is pointing at a kind of
indefinite extensibility in human natural linguistic
competence, since he refers in general to our
resources to define reals.
Moreover, Cantor seems to think that all reals are
definable, since any limitation on the cardinality of
the set of defining resources results for him in a
corresponding limitation on the cardinality of the
continuum.
This, I think, has little to do with the philosophy of
mathematics generally ascribed to him.
Again I would be grateful for any indication about
Cantor's letter.
Regards,
Laureano Luna
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