FOM: Rigidity
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Thu Feb 21 02:42:18 EST 2002
Randall Holmes, about a week ago, points out that a certain set theory
related to NF can prove that there ARE urelements, but that the urelements
are indistinguishable in the language of set theory, and (stronger claim)
that models of this theory admit of non-trivial automorphisms: any
permutation of the urelements of a model extends to an automorphism of the
model. He argues that this is not an undesirable feature in the theory,
suggesting that similar non-rigities are not necessarily bad in other
theories (e.g. of "arbitrary objects"). Comments and questions:
Comment 1: The point doesn't depend on the unusual choice of set
theory. It could be made in terms of the (closer to standard textbook) set
theory ZFC (in an urelement-tolerating version: e.g. as formulated in
Suppes's text) + an axiom saying there is a set of urelements of some
cardinality greater than 1. (This is an ARTIFICIAL example with an
"arbitrary" axiom-- Holmes would claim that his example involves a natural
set theory-- but maybe a more accessible one to some people.)
Comment 2: I've long felt that a mathematical theory whose models (at
least their "standard" models, when this distinction is applicable) DON'T
admit automorphisms seems to have a better intuitive claim to be theories
describing domains of "genuine" mathematical objects. Automorphisms in set
theories with urelements, however, seem o.k. I think this is because we
think of the the automorphisms as being there only because of the
limitation on the language: urelements don't have to have unique
set-theoretic properties: the urelements might, for example, be physical
objects, distinguished by their physical properties. PURE sets (= sets
containing no urelements in their transitive closures), in contrast, are
set-theoretic objects if they are anything: it WOULD, it seems to me, be
objectionable (& a reason for philosophical doubts as to the set theory's
claim to be describing genuine objects) if the intended model of a set
theory contained indistinguishable PURE sets.
QUESTION: It's an elementary exercise (maybe a good one for introducing
epsilon-induction to beginners) to show that pure sets can't be moved
around in the universe of ZF with foundation (any non-trivial automorphism
of a standard model must move some urelement). One of the nice features
of several of the "antifoundation" axioms considered in Aczel's monograph
"Non-Well-Founded Sets" is that they support similar results. Question for
Randall: what about NF and its cousins? Would their models admit
automorphisms that moved pure sets?
Comment: Rigidity (nonexistence of nontrivial automorphisms) of models
seems to me to be a very important notion from the standpoint of the
philosophy of mathematics. It's a notion that one WANTS to characterize by
saying that every object in the model is uniquely specified by in the
language of the theory, but that's not right if the syntax of the language
is standard First Order Logic. One can say (can't one?) that a model is
rigid iff every object in the domain has a unique description in the
INFINITARY version of the language (arbitrarily long conjunctions and
disjunctions; arbitrarily long sequences of quantifiers).
Bibliography: I connect this with the traditional metaphysical topic
of the "identity of indiscernibles." Two papers likely to appeal to
logicians & FoM-ers are
W.V. Quine, "Grades of discriminability," JOURNAL OF PHILOSOPHY 1976
(but I think I remember that the version in Quine's book "Theories and
Things" incorporates a correction), and
Randall Dipert. "The Mathematical Structure of the World: the World as
Graph," JOURNAL OF PHILOSOPHY, 1997 (this is an unusual paper, more
speculative than is fashionable-- the opening pages don't seem to have much
LOGICAL content, but automorphisms receive a fair bit of discussion later).
--
Allen Hazen
Philosophy Department
University of Melbourne
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