From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Numbers and sets
Organization: Boise State University
Date: Fri, 18 Dec 1992 21:45:00 GMT

In reading Jech's set theory book, I have noted that ZF- is used with
a different meaning than I have given it here.  The theory which I am
referring to as ZF- is ZFC - Foundation - Choice.  This theory has the
same consistency strength as ZFC or ZF; in fact, the well-founded sets
in ZF- make up a class model of ZF.  It should be noted that ZF-,
while it cannot define a notion of cardinal number of an arbitrary
set, can define a notion of "cardinal number of a well-founded set",
the notion needed for the interpretation of ZF.

ZF- embodies a notion of set which does not depend on the ordinals,
and it is capable of interpreting ZF and ZFC.  The only small point
which Zeleny wins is that ZF- cannot define canonical cardinal number
objects for itself (although it does define the relation of having the
same cardinality just as usual), but it can define the canonical
cardinal number objects for ZF.  Thus, it can be conceded to Mikhail
that the notion of cardinal number (but not the notion of
equinumerousness) in ZF depends on the prior development of the notion
of "well-founded set" (not the notion of "ordinal").

I'm not sure if I've noted an error here:  my definition of a stage as
a "universe" which contains all its proper subsets which are
"universes", where a "universe" is a transitive set which contains all
subsets of its elements, is incorrect.  All sets which have this
property are stages of the iterative hierarchy, but not all stages of
the iterative hierarchy have this property.

A correct definition of "stage of the iterative hierarchy" which does
not depend on the ordinals is as follows:

Definition: Let S be any set.  A set A is said to be "hierarchically
closed" relative to S if for each x which is an element of A such that
P(x) (the power set of x) belongs to S, P(x) also belongs to A, and
for each B which is a subset of A such that U[B] (the union of B)
belongs to S, U[B] is an element of A.  Let F[S] be the set of subsets
of S which are hierarchically closed relative to S.  Let H[S] be the
intersection of F[S]; the elements of H[S] are the elements of S which
belong to every hierarchically closed subset of S.  We then say that S
itself is a "stage" if S is the power set of an element of H[S] or the
union of a subset of H[S] (in this case, the subset can be taken to be
all of H[S]).

This definition is a precise way of asserting that the class of stages
of the iterative hierarchy is the smallest class which is closed under
the operation of taking power sets of its elements and the operation
of taking unions of its subclasses.  Sets specified by such inductive
definitions are _conveniently_ (emphasis directed at Mikhail) indexed
by the ordinals, but this is _not_ necessary.  A definition of the
ordinals can be obtained in the same way by substituting the successor
operation x+ = union of x and {x} for the power set operation.

There probably is a correct definition of "stage" similar to the
mistaken one given above; there is a similar definition of the
ordinals which does succeed.


-- 
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
Newsgroups: sci.math