NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/logic/2109

Text File | 1992-11-19 | 1.6 KB | 35 lines

Newsgroups: sci.logic
Path: sparky!uunet!stanford.edu!CSD-NewsHost.Stanford.EDU!Sunburn.Stanford.EDU!pratt
From: [e-mail] (Vaughan R. Pratt)
Subject: Re: recursive definitions and paradoxes
Message-ID: <[e-mail]>
Sender: [e-mail]
Organization: Computer Science Department,  Stanford University.
References: <[e-mail]>
Date: Thu, 19 Nov 1992 21:50:48 GMT
Lines: 23

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.
--
Vaughan Pratt              A fallacy is worth a thousand steps.