FOM: Survey/Constructibility/Issues

[Harvey Friedman]

This is a comment on the posting of Brown  Sun, 12 Mar 2000 20:49
concerning his survey.

I would like to call attention yet again to a point I have been making
about V = L (the axiom of constructibility). See my FOM postings

Sat, 12 Feb 2000 19:35
Tue Mar 07 11:33:47 2000

The principal point is that V = L is much better viewed as a clarification
than as an axiom.

If one is to evaluate V = L as an axiom in terms of how compelling its
truth is about the set theoretic universe taken in its full generality as
intended by Cantor, then that evaluation is likely to be poor. This is
reflected by the 0% response to V = L in the survey - and I answered the
survey.

However, V = L also serves the role of a

**pragmatic clarification**

in reaction to the unsettled state of set theory when taken in its full
generality as intended by Cantor. This unsettled state is such that
apparently very few people now expect a compelling solution to the
continuum hypothesis of the long awaited kind. The same can be said of many
many other well known set theoretic problems.

DIGRESSION: We could run a poll on what people expect to happen with the
continuum hypothesis and related problems, in terms of compelling solutions
of one sort or another. END.

As a pragmatic clarification, V = L works extremely well in many senses. E.g.,

1. It settles virtually all well known set theoretic problems.
2. It is demonstrably consistent relative to ZF.
3. Thanks to early work of Jensen, for many if not most applications, it
can be replaced by sensible combinatorial set theoretic consequences such
as diamond, box, etcetera.
4. It is an agreement to restrict the set theoretic universe in a logically
coherent and robust way.

Within the context of mathematical theories, the restriction of the
category of objects considered - in coherent and robust ways - is a time
honored major theme in modern mathematics. E.g., groups to countable groups
to finitely generated groups to finitely presented groups. This is normally
done when a theory proves to be overly general, and gets bogged down in
questions whose difficulties are tied up with the generality, rather than
with the original issues that the theory was originally designed to
address. The original issues normally lie in the concrete cases.

In this sense, agreement to work within the constructible universe of sets
would be a very natural move for the normal mathematics community. In fact,
I have no doubt that this would be the preferred solution of the normal
mathematics community to the very unsettled state of affairs in set theory,
if - and this is the big if - they cared about set theoretic mathematics.
And going this route would not require normal mathematicians to manipulate
constructibility technology in their daily work.

However, the fact is that they don't now - and never will - care anywhere
near enough about such things as the continuum hypothesis, Souslin's
hypothesis, measurability of PCA sets, etcetera, in order to get involved
in the relevant foundational issues. So these issues do not even get joined
for the normal mathematics community.

But the promise of Boolean relation theory is that deep foundational issues
must inevitably get joined, in connection with the introduction of long
well orderings and other related foreign technologies into normal
mathematics. And under this expected development, it does no good
whatsoever for the normal mathematics community to restrict to
constructible sets - or for that matter - to restrict mathematical objects
in any way whatsoever. The long well orderings and related technologies are
still demonstrably required.  As long as one is working with integers and
finite sets/functions in the integers, one is expected to be forced into
deep foundational issues. Absolutely no normal mathematician has ever
suggested a restriction on the integers and finite set/functions in the
integers, and that is not going to be viewed as a viable option by them in
order to hide from any impending invasion.