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From: lazarus@brahms.BERKELEY.EDU (Andrew J &)
Newsgroups: net.philosophy,net.math
Subject: Re: Ineffable numbers
Message-ID: <11077@ucbvax.BERKELEY.EDU>
Date: Mon, 25-Nov-85 15:17:20 EST
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Posted: Mon Nov 25 15:17:20 1985
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Xref: watmath net.philosophy:3230 net.math:2561

These so-called 'ineffable' numbers date back to one of the
turn-of-the-century paradoxes.  I think it's the Burali-Forti
paradox, but that course was many years ago and I don't remember.

Essentially it is easy to derive paradoxes involving these
objects.  The Cantorian construction posted previously is
quite cute, viz., 

1.  The effable (?!) numbers in [0,1] are countable since the
number of names is countable....

2.  Use Cantorian diagonalization to derive a number not
in the effable list....

3.  But this number is effable (contradiction)

However, a better question might be "So What".  Paradoxes
arising from circular definitions like effability are common.
I somehow recall the specific violation of set theory involved
is called "impermissible quantification".

andy