🚀 go-pugleaf

Re Gudeman:

Your criterion for an acceptable property is wrong.

The open sentence "x is not an element of x" is in fact true or false
for each x, in any reasonable set theory.  But it does not define a
set.  The reason it does not define a set is that _if it defined a
set_ there would be a problem with substituting some particular x (the
extension of this property) into the sentence.  There is in fact no
instance in either ZFC or NFU in which "~x E x" fails to be true or
false for some x; in ZFC, it is _always true_, and in NFU it is
sometimes true and sometimes false.

My appeal to particular set theories is merely to point out that this
situation is coherent and belies your criterion for a property to be a
"real" property.  Some perfectly well-defined open sentences which
have truth values no matter what is substituted into them cannot have
extensions or intensions.  Once again, there is no problem with
self-membership or non-self-membership holding of particular objects;
there simply is no "object" which is the extension of
"self-membership" or "non-self-membership".  And this does not mean
that the objects in the universe are not unambiguously divided between
the self-members and the non-self-members; there is no ambiguity, no
gray area between them.  But there cannot be a set of the
non-self-members.

-- 
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