[FOM] Finite sets: two references

[A.P. Hazen]

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