[FOM] Cantor on Richard's Paradox

laureano luna laureanoluna at yahoo.es
Mon Jul 2 11:20:03 EDT 2007

To my surprise I have found the following at

>Georg Cantor wrote in a letter to David Hilbert:
>(...)If Königs statement was "correct", according to
which all "finitely definable" real numbers form a
>collection of cardinal number aleph_0, this would
>imply the countability of the whole continuum; but
>this is obviously wrong. The question is now what
>error the alleged proof of his wrong theorem is
>based upon. The error (which also appears in the
>note of a Mr. Richard in the last issue of the Acta
>mathematica, which Mr. Poincaré emphasizes in the
>last issue of the Revue de Métaphysique et de
>Morale) is, in my opinion, the following: It is
>assumed that the system {B} of notions B, which have
>to be used for the definition of individual >numbers,
is at most countably infinite. This >assumption "must
be in error" because otherwise we >would have the
wrong theorem: "the continuum of >numbers has
cardinality aleph_0". 

My questions are:

Cantor seemingly believed the set of possible
definitions of reals was not countable: how was this

Does someone has more information about that letter?


Laureano Luna

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