[FOM] A Defence of Set Theory as Foundations

[Nik Weaver]

Broadly speaking, one can attempt to justify the axioms of set
theory in two different ways.  In one approach one tries to
identify sets with some well-defined class of independently
existing entities --- sets are Platonic forms, sets are "ideas",
or what have you --- and then argue that on this basis the axioms
are evident.  To me, approaches of this type have a pre-scientific
flavor to them.  So the first problem one faces when defending a
view of this sort is to make a case that the claim (e.g., "sets are
ideas") is even meaningful.  The second problem is that it seems
that if any approach of this type were to succeed, it would actually
justify full comprehension. That is, this kind of platonic or
"realistic" stance seems aimed at justifying not axiomatic set
theory, but rather naive set theory, which we know is inconsistent.

In the other approach, constructivism in a broad sense, one
attempts to construct, or imagine constructing, a domain which is
supposed to play the role of a universe of sets.  The natural way
to do this is to build the domain up in stages, which leads to an
"iterative" conception of sets.  The problem with this view is that
it fails to support the power set axiom, since we have no clear
picture of how to construct the power set of an infinite set.  In
fact, power sets of infinite sets are fundamently circular in a way
which I discuss in my paper "Mathematical conceptualism" --- see the
messages I posted last month for references --- suggesting that one
has no hope of justifying power sets on a constructivist basis.

The iterative conception which is supposed to justify ZFC is thus
an incoherent combination of two contradictory set concepts.  On
the one hand, we imagine building up the universe of sets in an
iterative fashion, but on the other hand power sets are permitted
to simply appear fully formed, without being constructed.  This
leads to (at least) two serious obsurities:

1.  Are we regarding sets platonically or constructively?  The
admission of power sets is inconsistent with the latter, but the
idea that we are avoiding the classical paradoxes by building up
the universe iteratively is inconsistent with the former.

2.  What brings the set-theoretic universe into being?  The concept
of a dynamical process which does not take place over time is not
coherent.

In my paper "Analysis in J_2" I show how mainstream mathematics (up
to general topology, measure theory, and functional analysis) can be
developed cleanly and elegantly in a system which is constructively
legitimate.  Thus, it is possible to take a principled approach to
foundations, accept that one cannot form power sets, yet still retain
core mathematics in essentially its classical form.

The most substantial objection to this position is that there exist
important mainstream mathematical theorems which it cannot support.
Perhaps the most compelling example is Kruskal's theorem.  However,
I show in my paper "Predicativity beyond Gamma_0" that this objection
is not valid; indeed one can prove Kruskal's theorem predicatively.
I have not yet received any specific criticism of the analysis given
in this paper.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu