NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/logic/2124

Internet Message Format | 1992-11-20 | 2.5 KB

Path: sparky!uunet!destroyer!ncar!noao!arizona!gudeman
From: [e-mail] (David Gudeman)
Newsgroups: sci.logic
Subject: Re: recursive definitions and paradoxes
Message-ID: <[e-mail]>
Date: 20 Nov 92 17:46:33 GMT
Organization: U of Arizona CS Dept, Tucson
Lines: 41

In article  <[e-mail]> Vaughan R. Pratt writes:
]In article <[e-mail]> [e-mail] (David Gudeman) writes:
]>But consider the paradoxical definitions
]>
]>  S := {{},R}
]>  R := {x E S : ~(x E x)}                                (8)
]>
]>where the definition of R has exactly the same form as the definition of
]>Russell's paradoxical set (replacing S with the universal set).  But
]>here, S is a set with only two elements!  Given such a clear example of
]>Russell's paradox by quantifying over a finite set, I don't see how
]>anyone can still claim that Russell's paradox is somehow caused by a
]>universal set that is "too big".  Size has nothing to do with it, the
]>only consideration is how it is defined.  Clearly in this example the
]>problem is an ill-founded mutual recursion between the two set
]>definitions.
]
]Russell's paradox denies the existence of the paradoxical set of all
]sets.  Denial of an existential is a universal, and the universal
]statement asserted by Russell's paradox is that every set has a
]nonmember.  The universal set is "too big" by one element, having no
]nonmember.  From that perspective size has everything to do with it.

Your argument, depending as it does on universals, cannot account for
the finite paradox I present above (nor can it account for a host of
other logical and "linguistic" paradoxes).  Therefore, if you insist
on this explanation of Russel's paradox, you require two different
explanations for two very similar paradoxes.  I can explain both of
them at once.  Doesn't this give my explanation more credibility?

My explanation of the paradoxes (I claim) applies universally to _all_
the common paradoxes of logic --including the so-called linguistic
paradoxes.  I've actually gone through a large subset of them to
verify this.  In each case, I could find no way to write the paradox
without some form of recursion, and in each case, my axiom of
definition (or its equivalent) would render the paradox harmless.
Isn't this "unified paradox theory" better than a set of theories that
requires different explanations for different paradoxes?
--
                    David Gudeman
[e-mail]