🚀 go-pugleaf

It should be made clear that the identity of the One True Set Theory
is not the issue in the argument about "quantification over
everything".  Zeleny's position makes no sense from the standpoint of
the very Cantorian set theory he professes to regard as the Truth (and
which I agree is an accurate description of part of the whole).

When we quantify, we do not quantify over a set.  The statement "for
all x, P" means that P is true no matter what x is (there may be
implicit restrictions in given disciplines; in ZFC we are, for
example, talking only about well-founded sets -- but note that
well-founded sets do NOT make up a set), not "P is true whenever x is
taken from a certain collection".  Moreover, unrestricted quantifiers
can occur in conditions which "define" sets in ZFC; if we restrict
ourselves to defining sets using conditions with quantifiers
restricted to sets, we cannot prove the existence of aleph-omega (we
are essentially working in type theory).

As Zeleny has said himself, there is no completed totality of all the
sets of ZFC; this is an Absolute Infinite totality, in Cantor's sense.
How does he square this with his insistence that there must be such a
totality (in a a higher type, apparently) to quantify over?  This
seems to arise from his belief that ZFC "has semantics" in the sense
that the intended interpretation of ZFC is a model of ZFC.  But a
model of theory T is a _set_; the intended interpretation of ZFC
describes an Absolute Infinite totality which cannot be construed as a
set! 

If we admit his classes, super-classes, and so forth, he is doing
omega-order ZFC.  He has not gained anything by this regress; I'm
looking for his model of omega-order ZFC...



-- 
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