FOM: f.o.m. and the concept of set

[wtait@ix.netcom.com]

A remark about the relationship between foundations and set theory: An 
old-fashioned conception of foundations is, roughly, that it is concerned 
with finding the right axioms.  Set theory then is the locus of 2 
problems in f.o.m.: CH and Transfinite numbers. In the case of 
transfinite numbers, the task of finding the right axioms is never 
completable, since whatever axioms S we accept, we should be willing to 
take as a further axiom the existence of a model for S. (Zermelo 1930 
states this principle.) This problem is tied to set theory in at least 2 
ways: First, with the system of transfinite numbers is associated the 
cumulative hierarchy of sets iterated along the system of transfinite 
numbers. It is natural to think of the axioms that we might introduce for 
the existence of new numbers as formulated in the language of this 
hierarchy of sets. Second, the question of what numbers we admit is 
equivalent to the question of what totalities of numbers we admit as sets 
(since to admit it as a set is to admit its lub as a number and 
conversely.) 

I want to emphasize that this is not an argument for `set theoretic 
foundations' over `category theoretical foundations'. I simply want to 
recall a fact, of which everyone is aware, but which has sometimes been 
lost in recent discussions, viz. that the concept of set is intimately 
implicated in leading problems in (old-fashioned) f.o.m. The issue is NOT 
whether the language of categories could replace the language of set 
theory in formulating axioms e.g. about transfinite numbers. 

I do wonder whether the very extensive debate on the list over set 
theoretic vs category theoretic foundations is something more than a 
debate about rhetoric, about which of equivalent languages one should use 
to, e.g., write textbooks on analysis for school children or whomever.

Bill Tait