[FOM] 600: Removing Deep Pathology 1

[Harvey Friedman]

On Wed, Aug 26, 2015 at 3:00 PM, Mitchell Spector <spector at alum.mit.edu> wrote:
> Are these "pathological" sets still "pathological" in L, even though there's
> an explicit construction for them there?
>
> If you answer "Yes", then the "pathology" isn't really based on
> non-constructivity after all.
>
> If you answer "No", then you have to be willing to concede that, for
> example, the Banach-Tarski paradox isn't bizarre in L.

I am talking only about what I call "deep pathology" where there is an
absence of provable definability.

There are various levels of pathology of great interest, all the way
down perhaps to the problematic idea of a pathological finite set of
integers.

What you are talking about here is, for example, an explicit
construction of a well ordering of the constructible reals. There is a
formula which, provably in ZFC, defines a well ordering of the
constructible reals. However, it can be argued that it is highly
pathological.

I am happy to view this as highly pathological, but it falls short of
the deeply pathological that I am talking about.

Incidentally, this raises the issue of the mathematical naturalness of
L, or fragments of L like the constructible reals.

The most mathematically natural description of  (all of) L that I know
of is in class theory:

L is the least transitive proper class X where (X,epsilon) satisfies

*any set {x in X: phi(x) holds in (X,epsilon)} lies in X*

Here phi can of course have parameters from X.

Two objections can be made about this definition. One is that it
relies on the very general notion of set represented by the cumulative
hierarchy - unknown to normal mathematicians. The second is that it
relies on first order definability - unknown to normal mathematicians.

Let me address the second objection. We know through our experience
with finite axiomatizability that we can in principal provide a finite
list of instances. I worked on this issue in the context where one is
starting from scratch, not from V. See
http://www.cs.nyu.edu/pipermail/fom/2015-August/018873.html So 4
quantifier, two parameters suffice even in impoverished situations. SO
HERE we can expect something much better. Namely, one should be able
to enumerate a very small number of natural instances of {x in X:
phi(x) holds in (X,epsilon)} that work, and are very satisfying. I
tried to do that for the impoverished setting of
http://www.cs.nyu.edu/pipermail/fom/2015-August/018873.html and the
list got nasty. But here it probably won't be nearly as nasty.

With regard to the first objection, that we are relying on the general
concept of set. There are several ways to formulate challenges here.

1. Work in ZC. Now give such a mathematical definition of L What kind
of L? It could be L intersect V(omega + omega).

2. Work in ZFC as usual. Give a mathematical definition of the set of
constructible subsets of omega.

Here is something I am planning to discuss on the Deep Pathology series.

What is a reasonable set of reals?

I like to use Borel, and that is very rich and satisfying. However, it
may rightly be viewed as overly restrictive. And I don't want to use
some technical thing in descriptive set theory.

I came up with the following nice attribute. EVERY BOREL IMAGE HAS THE
BAIRE PROPERTY.

What I am getting at is this. You can come up with deeply pathological
sets of reals with the Biare property, because they merely are nowhere
dense. This is in a sense defeated by the above. And why not insist on
both of these:

EVERY BOREL IMAGE HAS THE BAIRE PROPERTY AND IS LEBESGUE MEASURABLE.

Remember the purpose. I want to take some bad spaces in math that
everybody is talking about, and NICELY define the good part. So, you
can take the Borel part and that works terrifically in a lot of
situations. You can also take the part according to the above, and get
the same nice thing.

Harvey Friedman