🚀 go-pugleaf

In article <25916@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
>In article  <1992Nov04.233228.16942@Cookie.secapl.com> Frank Adams writes:
>]In article <24780@optima.cs.arizona.edu> gudeman@cs.arizona.edu (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.

Dare I concur?

>-- 
>					David Gudeman
>gudeman@cs.arizona.edu



-- 
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