🚀 go-pugleaf

In article <1992Nov19.215048.26539@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <26788@optima.cs.arizona.edu> gudeman@cs.arizona.edu (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.

This is inaccurate.  Russell's paradox proves nothing except that
there is no set which has exactly the sets which are not elements of
themselves as members.  If the Axiom of Separation is added to one's
theory, the paradox has the consequence indicated.  In theories where
the Axiom of Separation does not hold (like NFU) the paradox has
different consequences.

Separation = for any set A and each formula P, {x E A | P} exists.


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu