[FOM] Finite sets: two references
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Thu Feb 23 21:16:56 EST 2006
Harvey Friedman, in a number of recent posts, has argued that ZFC is
a **natural** analogical extension to infinite sets of an easily
understood and **natural** (I'm using **natural** here as shorthand
for something -- or some things --that can be discussed at length)
theory of finite sets, and has given us some logical comments on the
nature of the analogical extension involved (what "transfers" to the
infinite case). Two bibliographical comments.
(1) The only previous attempt I know at saying something
substantive and precise about the nature of the analogical extension
involved in going from our experience with the finite to a theory of
infinite structures is Shaughan Lavine's "Understanding the
Infinite." The approach is very different from Harvey's, but maybe
not incompatible (they may be looking at different aspects of the
problem!). I won't say more about it now, just that it is very
interesting (my personal nomination for "most interesting
contribution to the philosophy of mathematics" in the decade of its
publication), and I recommend it to anyone interested in how there
can be an **analogy** between finite and infinite.
(2) In the discussion a few people have raised a familiar worry
(familiar at least in the philosophical literature) of whether our
"ordinary" understanding of **sets** covers the null set or allows
unit sets to be distinct from their members. This issue, it seems to
me, is simply a distraction from the point of view of Foundations
of Mathematics (though, maybe, worth some discussion in connection
with the METAPHYSICAL side of the philosophy of mathematics). It's a
distraction because (as long as WE -- YOU and I, for example -- are
willing to assume the existence of at least two urelements to get
things started) a theory of plurimembered sets (= sets each of which
has two or more members) can be constructed which has the same
MATHEMATICAL content as the usual set theory. This can be shown by
showing that the usual theory can be interpreted, formally, in the
plurimembrate theory. (After which, since allowing null and unit
sets is MUCH more convenient when it comes to DOING set theory....)
... ... I argued this a long time ago, giving the
interpretation for the simple case of simple type theory (obviously
the interpretation would be more complicated for ZFC, and there may
even be delicate issues if you try to run the argument for versions
of set theory without foundation -- but I felt the simple formal
example was enough given the elementariness of the philosophical
argument), in an article, "Small sets," in "Philosophical Studies,"
v. 63 (1991), pp. 119-123. Anyone wishing to raise the "small set"
problem on a Foundations of Mathematics forum should, it seems to me,
EITHER argue that we don't have a a clear understanding of
plurimembrate sets (and not waste time on the nulland unit cases)
OR argue (this seems harder to me) that there is something very
vicious in the way unit and null sets can be coded by plurimembrate
sets.
I'll shut up now.
--
Allen Hazen
Philosophy Department
University of Melbourne
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