NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/logic/1935

Internet Message Format | 1992-11-06 | 1.7 KB

Path: sparky!uunet!destroyer!ncar!noao!arizona!gudeman
From: [e-mail] (David Gudeman)
Newsgroups: sci.logic
Subject: Re: Russell's Paradox
Message-ID: <[e-mail]>
Date: 5 Nov 92 20:22:02 GMT
Organization: U of Arizona CS Dept, Tucson
Lines: 26

In article  <[e-mail]> Frank Adams writes:
]In article <[e-mail]> [e-mail] (David Gudeman) writes:
]>This (or so I claim) is the single, simple, obvious cause of Russel's
]>paradox.  In particular, R = {x : x not elem x} is a definition of
]>type (2) that contains an implicit reference to R because x quantifies
]>over a set that contains R.  (Or you could show the recursion in the
]>definition of elem as I did before)...
]
]It appears to me that you are complaining not about elementhood being a
]proposition, but about the unrestricted range of the quantification in the
]axiom of comprehension.  Note that your complaint about the implicit
]reference to R is just as applicable to R = {x : x = x}, which gives the
]universal set without using elementhood.

No, I am not complaining about elementhood being a proposition, I am
complaining about a syntactic circularity begin treated as though it
had semantic significance.  Neither am I concerned about the
unrestricted range of the quantification.  As the set {x : x = x}
clearly shows, there is no inconsistency inherent in either
self-membership or impredicativity.  I claim that {x : x = x} is a
perfectly reasonable set and that the mere fact that ZF proves the
opposite is enough to show that ZF is not an adequate axiomatization
of set theory.
--
                    David Gudeman
[e-mail]