[FOM] iterative conception/cumulative hierarchy

Christopher Menzel cmenzel at tamu.edu
Sat Feb 25 17:12:25 EST 2012

Am Feb 25, 2012 um 8:52 PM schrieb Michael Lee Finney:
> Thank you.
> I have seen that paper before, but its been a while. I downloaded it
> again (another $10, arrggghhh!) and reread it.
> However, that doesn't change my opinion. Essentially, I reject both
> the doctrine of limitation of size and the doctrine of the hierarchy.

But this is all overly-doctrinaire, isn't it?  Unless one is convinced there is simply the One True Concept of Set, these aren't doctrines, they are, in the one case, a working principle to guide the formation of set theoretic systems and, in the other, a particular mathematical structure. The principle evolved out of a growing awareness of a notion of "over-largeness" distinguishing problematic collections (Cantor himself was fully aware of it) that eventually took a mathematically definite form (one among others) in the proper classes of the cumulative hierarchy. You may well not *prefer* for one reason or another the conception of set, and the concomitant distinction between set and class, reflected in the cumulative hierarchy, but you can't "reject" the hierarchy itself any more than you can "reject" the natural number structure; it is as legitimate and objective a mathematical structure as any group or category -- as is the non-well-founded universe of AFA, the structures determined by NF-style set theories, etc. One structure might reflect a more suitable conception of set than another for some purposes but all are worthy of mathematical study.

> I consider these to be emotional crutches that many people trained in
> the 1800's needed to deal with infinities. And, because of the paradoxes,
> they sort of stuck

The effectiveness of the cumulative hierarchy in providing a compelling (not to say unique) explanation of the set theoretic paradoxes no doubt contributed to its persistence.  But your armchair speculation that the cumulative hierarchy emerged out of the emotional neediness of the historical founders of modern set theory is patently ridiculous.

> Once Peter Aczel did his work on non-well-founded sets, I cannot see
> any justifiable reason to accept the axiom of regularity.

Again, these lines in the sand are just not really to the point. If you wish to study, say, inner models of ZF within the cumulative hierarchy, regularity will be among the axioms you adopt. If that's not your cup of tea or you find regularity unsuited to your purposes, don't use it.

Chris Menzel

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