From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Numbers and sets
Organization: Boise State University
Date: Thu, 17 Dec 1992 23:59:06 GMT

In article <1992Dec15.214124.7444@guinness.idbsu.edu> holmes@opal.idbsu.edu (Randall Holmes) writes:
>I repent!  I repent!  No >>>>>>>>>>>>>>>>>!
>
>My final answer to Zeleny is that I accept his point in a modified
>form (which for him may not count as acceptance at all.)  I think that
>the sets of Zermelo set theory and its extensions (with the Axiom of
>Foundation) are best understood as being related to isomorphism types
>of well-founded extensional relations.  Insofar as ordinals (best
>understood as being related to isomorphism types of well-orderings)
>are a prominent and structurally significant special case of this kind
>of relation, I agree that the ordinals are essential to understanding
>ZF.  This is sharpened when ZF is extended to ZFC by the addition of
>the axiom of choice.  The particular approach of ZF makes it
>impossible to choose canonical objects to serve as cardinal numbers
>without using foundation or choice.  The concept of cardinal number
>(although not the relations of having less or greater cardinality)
>depends on foundation or choice for its expression.
>
>But the axioms which mediate this relation with the ordinals (or more
>generally with well-founded extensional relations) are strictly
>foundation and choice.  ZF- is still a powerful set theory (it is no
>weaker than ZFC in terms of consistency strength).  Its axioms of
>separation and replacement embody a concept of _set_ (avoiding paradox
>by application of the "limitation of size" doctrine) which is not
>dependent on the ordinals.  ZF- is not studied much that I know of,
>but the situation is the same in ZFA, in which extensionality is
>weakened to allow atoms (I believe that Zermelo's original theory
>allowed atoms, which argues against Zeleny's claim about the history
>of the set idea!) without choice, and this theory is studied; it was
>the first theory for which models without choice were constructed.
>The iterative hierarchy is a powerful working hypothesis, but it is
>not a prerequisite for the modern concept of set, but a refinement
>thereof.  Notice that it _does_ appear in ZFA or ZF- as the structure
>of _part_ of the universe.
>
>
>By the way, Aczel's AFA is just as good as foundation; ZF- + AFA does
>have well-defined canonical cardinal numbers.
>
>A definition of "stage of the iterative hierarchy":  let a "universe"
>be a set which contains all elements of its elements and all subsets
>of its elements.  Then a "stage of the iterative hierarchy" is a
>universe which contains all its proper subsets which are universes.
>The correctness of this definition is left as an exercise, but note
>that it does not depend on the ordinals!

Alas, this definition is incorrect.  All sets which have this property
are stages, but not all stages have this property.  I was working by
analogy with the definition of "ordinal" as "transitive set which
contains all its transitive proper subsets"; I'm sure there is a
refinement of this approach which does work, and I have another
definition on tap which is correct.

>
>-- 
>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


-- 
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
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