[FOM] What do you lose if you ditch Powerset?

[Harvey Friedman]

Reply to Chow.

On 11/14/03 1:46 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> Sorry if this is a dumb question with a well-known answer, but...

This matter has been considered by at least me for several decades.
> 
> What sorts of theorems are lost if we delete from (say) ZFC the assumption
> that every infinite set has a powerset?
> 
> Perhaps more precisely, what sorts of theorems of "classical" mathematics
> are there that (1) do not, on the surface at least, require the existence
> of powersets of infinite sets for one to make sense of their statements,
> but (2) cannot be proved without the powerset axiom or something similar?

Let ZFC\P be the result of dropping the power set axiom from ZFC.

All presently known "natural" mathematical examples of sentences provable in
ZFC but not in ZFC\P fall into two groups.

1. Examples over several decades involving Borel measurable functions on
Polish spaces (complete separable metric spaces).

2. Examples from Boolean relation theory that don't quite exist yet, because
my priority has always been on independence from ZFC.

3. Under a moderately strict notion of "classical" mathematics, there are no
known examples. 

Since 2 does not yet exist, we concentrate on 1.

Borel measurable functions, even on R, *superficially* use the power set for
their formulation, since R cannot be proved to exist *as a set* in ZFC\P.
However, there is absolutely no need to have R as a set in order to discuss
Borel measurable functions on R. In fact, the whole theory of Borel
measurable functions on Polish spaces has a completely natural development
without using R as a set. One uses, instead, "being a real number", and uses
countable well founded trees in order to present Borel measurable functions
in a totally natural way.

The main examples are

A. Borel diagonalization. In general terms, this says that in many contexts,
a Borel measurable function satisfying some invariance conditions, always
maps some argument to a value that is "simpler" than that argument. The
original such statement is: every Borel measurable function from infinite
sequences of real numbers into real numbers, that depends only on the range
of the sequence, must map some sequence to a coordinate of that sequence.
These examples are provable in ZFC but not in ZFC\P. Other examples
necessarily use uncountably many iterations of the power set operation -
much stronger than ZFC\P, but within ZFC.

B. Borel determinacy (in infinite two person zero sum games). It is
necessary and sufficient to use uncountably many iterations of the power set
operation to prove Borel determinacy.

C. Two sided Borel selection. Closely related to B. Every Borel set in the
plane contains or is disjoint from the graph of a Borel function from R into
R. It is necessary and sufficient to use uncountably many iterations of the
power set operation to prove two sided Borel selection.

D. Borel selection theory. This is just a few years old. Various statements
published by functional analysts at U. Paris VII, when specialized to Borel
functions, can be proved in ZFC but not in ZFC\P and, in fact, not in
fragments of ZFC with only countably many iterations of the power set axiom.

The following reference discusses this matter, as well as other related
independence results from ZFC and beyond, and has appropriate references to
all of the above.

Harvey Friedman, Rademacher Lectures,
http://www.mathpreprints.com/math/Preprint/show/index.htt

Also see 

Harvey Friedman, Borel Selection,
http://www.mathpreprints.com/math/Preprint/show/index.htt

Harvey Friedman