FOM: Re: axiom of infinity

[A.P. Hazen]

I'm not sure the status of the axiom of infinity is an ISSUE in the
foundations of mathematics.  Maybe in metaphysics/philosophical
logic/philosophy of language, but not for the foundations of mathematics.
Quine's pragmatic approach to this (to give it a name!) issue will probably
be attractive to most FoM workers: in formulating a foundational system for
(classical) mathematics or a fragment of it, you will want to postulate the
existence of something set-ish.  You can call them PROPERTIES (and not
postulate extensionality) if you want, but it's simpler and neater and so
on to postulate extensionality and call them SETS.
This is o.k., because (probably!) adding the extensionality postulate isn't
really begging any questions, and won't smuggle undeclared mathematical
power into your system.  This is because-- not in all imaginable contexts,
but under fairly general assumptions-- a system with extensionality added
can be interpreted in the corresponding system without it, and so doesn't
add serious deductive power or serious risk of inconsistency.  The idea can
be explained simply for Timple Type Theory: define a new equality as
coextensiveness for 'sets' of individuals, and restrict your quantifiers at
type n+1 to range only over type n+1 sets containing as members any type n
sets 'equal' to sets they alread have as members.
Something like this goes back to "Principia Mathematica."  Details, for a
specifically MODAL version of type theory can be found in Daniel Gallin's
"Intensional and Higher-Order Modal Logic" (North-Holland, 1975).  Robin
Gandy published papers on it in the JSL in 1956 and (I think) 1959-- if I
recall correctly, the first treated simple type theory and the second
Zermelo set theory.
The problem isn't QUITE trivial.  Without extensionality, for example, the
axioms of replacement and of collection are no longer equivalent.  And if
one were to formulate mathematics in an intensional language (perhaps like
the "Epistemic Mathematics" proposed by Nicholas Goodman and Stewart
Shapiro) the issue would have to be re-opened.
But my general prejudice is that the status of extensionality-- much like
philosophical cavils at the null set or at unit sets-- is a matter that MAY
be of some metaphysical interest, but has little relevance for mathematics,
or even for the  set-theoretic foundations of mathematics.
--
Allen Hazen
Philosophy Department
University of Melbourne
Interests: Curmudgeonry and Bibliography