[FOM] predicativity

[Roger Bishop Jones]

On Thursday 22 September 2005  2:52 am, Nik Weaver wrote:

> I gave a more principled defense in my last message and
> this seems to have been ignored.  The point is that
> impredicative (circular) definitions only make sense against a
> background of a well-defined preexisting universe of sets; if
> one does not believe there is such a thing, it becomes very
> hard to see why impredicative definitions should be allowed.
> On the other hand, one can argue that constructions of length
> omega are meaningful because it is possible to form a concrete
> mental image of them.  Combining these two ideas --- there is
> no preexisting universe of sets, and valid constructions must
> be concretely imaginable --- gives rise to the conceptualist
> view that I am defending.  Surely this is a better defense
> than simply saying "not as strong, hence not as dubious".

It seems to me that you are looking, not for a defence of
predicativity, since few people would argue against your
freedom to study predicative systems, but a critique of
classical set theory, since you seem to want people to
abandon set theory in favour of predicative foundations.

Your argument here against set theory is not convincing to me.
The notion of a pure well-founded collection seems to me
is as intuitively clear and simple as one can hope for in
a foundational ontology.

Even if we have doubts about whether it is coherent to
talk of the collection of all such sets, (which I do)
classical set theory can be interpreted in initial
segments of the cumulative hierarchy, obviating the
need for completion of the "construction".

It is inevitable that a foundation be incomplete, and
in some sense less well defined than any domain for
which it suffices to provide a foundation.
Anything less risky than set theory (which has proven
to be not at all risky) will of course be less complete
even in its arithmetic conclusions.

Your other arguments against set theory seem to
me less persuasive even than this one, and my reactions
to them follow.

Is the continuum hypothesis important to mathematics?
If it is not, then its undecidability in set theory is
not a problem, if it is important then how does abandoning set
theory in favour of a weaker foundation improve matters?
Given the inevitability of incompleteness how can examples
of incompleteness ever argue in favour of weak rather than
strong theories?
Does it progress mathematics to chose languages in which
difficult problems are not expressible; do the problems
then disappear?

Your observations about poor fit with mathematical practice,
seem simply to amount to the claim that most mathematics
concerns only small sets.
Obviously set theory itself, which surely is mathematics,
is a counterexample.
But in any case, why is it a problem if a foundation
provides more than most mathematics requires?

The Skolem "paradox", if a problem, is a problem with
first order logic, not with set theory.
Non standard models are of course just as much a feature
of first order arithmetic as of first order set theory.
If you don't like countable models of set theory, then
do set theory in second order logic
(with "standard semantics" of course).

It seems to me very strange to cite "the classical
paradoxes" (i.e. the paradoxes in naive set theory)
as serious concerns for axiomatic set theory.
In fact your concern here seems to be with the semantic
intuitions underlying the axiomatic set theories,
but the story about the iterative construction of the
cumulative hierarchy is now very old and venerable.
If you want to argue against it then you will need
to do better than speak as if there is no articulated
account of an intended interpretation of set theory.

I'm interested in critiques of set theory as a foundation
system, but I'm afraid yours fail to offer much food
for thought, which is of course, no reason why you should
not study and promote the merits of predicative systems.

Roger Jones