[FOM] Cantor on Richard's Paradox
Dennis E. Hamilton
dennis.hamilton at acm.org
Mon Jul 2 15:38:42 EDT 2007
I would think that it is the reverse. I believe that Cantor is clear that
the reals are not countable, and that the set of all possible (finite)
definitions of them does not embrace all of them (being countable).
Thinking that the reals are all finitely definable "would imply the
countability of the whole continuum" and hence is obviously wrong.
That seems to be how the letter should be read.
I think the discussion of notions at the end of the passage is consistent,
although I can see how that is easily read differently. I think the tricky
part is "definition of individual numbers."
- Dennis
-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of
laureano luna
Sent: Monday, July 02, 2007 08:20
To: fom at cs.nyu.edu
Subject: [FOM] Cantor on Richard's Paradox
To my surprise I have found the following at
http://allrss.com/wikipedia.php?title=Jules_Richard#Richard.27s_Paradox:
>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 error [...] 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
possible?
[ ... ]
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