🚀 go-pugleaf

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.