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Allen Hazen a.hazen at
Mon May 29 09:45:49 EDT 2000

A couple of remarks to Victor Sazonov's posting.
   (i)  The description of Anti-Foundation axioms as of interest more for
their applications (to, eg, analysis of "hyperlink" networks) than
mathematically seems exactly right.  Aczel's AFA (from his CSLI monograph
"Non Well Founded Sets") and its closer cousins (eg Scott's axiom,
discussed in an Appendix to the monograph) allow natural treatments in
applications like the theory of semantic self reference, but (and maybe
this is part of their attraction?) don't require or involve new
mathematical PRINCIPLES in the way that, eg, large cardinal axioms do.  A
non-wellfounded set, on either of these approaches, is characterized by the
pattern of the membership relation in its transitive closure.  (Scott's
axiom-- Aczel's is perhaps nicer, but is a bit more complicated to
remember-- identifies two non-wf sets iff the "graphs" of the membership
relation over the transitive closures of their unit sets are isomorphic.)
Further (since this is provable in ZF without appeal to either foundation
or antifoundation), the transitive closure of a set always exists, and
(Hartog's Theorem and AC) is of the same cardinality as some Well Founded
set.  So every non-wf set has a "picture" in the well-founded hierarchy: a
set of ordered pairs exhibiting the same isomorphism type as the membership
relation over the tr. cl. of the given non-wf set.  So ZFC+AFA is
interpretable in plain ZFC, so AFA is not a new axiom for mathematics.
  (i and 1/2)  It seems to me that the real significance of the AFA is that
it helps knock the "Iterative Conception of Set" off its metaphysical
pedestal.  Between about 1970 and 1990, the Iterative Conception was, at
least among philosophers, often taken in a very metaphysical way: sets
>>presuppose<<, or are in some quasi-causal way >>generated by<<, their
members.  (A particularly clear statement of this metaphysical
interpretation of the iterative conception is in the text-- the
"structuralist" tendencies of the appendix are another matter-- of David
Lewis's little book "Parts of Classes.")  The graph-theoretic inspiration
of AFA has, it seems to me, helped make alternative metaphysical views
about set theory seem more plausible.
		...Which is not to say the Iterative Conception isn't
foundationally helpful.  The infinite, topless, cone is still the best way
of visualizing the universe of ZFC; AFA just adds a fringe along the edge.
And, of course, on the other side, thinking of non-wf sets, and
applications using them, may help motivate mathematically new set theory.
  (ii) About what to call people who accept some constructivist philosophy:
I like "believing."  So we can call someone like the late Abraham Robinson,
who happily used set-theoretic methods while proclaiming his constructivism
or formalism, a "practicing, though not a believing, platonist."

Allen Hazen
Lecturer in Philosophy
University of Melbourne

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