[FOM] blurry pictures/incompleteness

[Harvey Friedman]

This is a copy of an email posted to another email list to Peter Koellner.

Peter Koellner wrote:

In those comments I go through the arguments one by one and argue that
they do not support the contention that CH is indefinite. (This is
largely done by comparing, in each case, the situation with that in
arithmetic, where Sol would not say that the statements are
indefinite.) The conclusion that I come to is this: Once these
arguments are seen to be wanting what is left is the claim that the
powerset of the reals (or any infinite domain) is not a clear notion.

*************
I certainly want to get a chance to look at the stuff you wrote about this,
but a thought occurred to me as I read the above.

There is a big difference between the power set of N and N itself. Namely,
there are a number of natural ways to restrict the power set of N so that
it does the overwhelming majority of work that mathematicians expect it to
do. This is not the case for N. Another way of saying this is: N is
naturally finitely generated, whereas the power set of N is not finitely
generated at all. Restrictions of N are not closed under successor.
Restrictions of the power set of N are closed under all of the
mathematically normal operations mathematicians use or are concerned with.

I wonder what you think of the Tony Martin quote at the top of Sol's paper?

Let me take the time here to encapsulate my own views on "clarity" or
"definiteness" or whatnot.

I believe in a sliding scale. I don't know how to assign numbers to this
sliding scale. But I am essentially perfectly comfortable with my mental
picture of n things, where n is pretty darn small. I divide into groups. I
can see clearly 8 as two groups of 4 where the groups of 4 are very clear.
I can do more, but at some point I am going to cheat just a little bit, and
simply make two copies of something I can see clearly. I no longer see the
totality so clearly as just one unit the way I could see 4. And so forth.

As I have to deal with bigger finite numbers, things start to blur in my
mind - in terms of these actual mental pictures. Not descriptions of how to
build mental pictures. But actual mental pictures.

With N, the the actual mental picture isn't really there in the strong
sense that it was for small numbers. There is just an idea of how to
continue one by one. A certain kind of flaw already arises with N.

Yet N is pretty darn good. At power set of N, it is significantly worse.
There is only the idea of what features a subset of N has to have in order
to be a subset of N. This is obviously somewhat clear, but in a different
league altogether than N. With N I am conscious of my attempt to grasp the
whole thing, with one quick pass to the right with my imaginary eyes. But
with the power set of N, in an attempt to grasp the whole thing - well I
don't move my eyes, and instead contemplate simply the idea of being a
subset of N, with no generation process. Just an "anything is OK" idea
(full comprehension). With power set of power set of N, I see another major
degrading of the mental picture. And so forth.

This happens to parallel mathematical incompleteness - is the parallel
something coincidental or is it fundamental?

That is, the higher you go, the easier it is to find perfectly natural
mathematical incompleteness. By far the hardest is Pi01 so far. But there
should be perfectly natural incompleteness in say 2^100 taken into account
absurd lengths of proofs.

What really allows us to work so effectively with so many abstract entities
is that we really don't have to know much about what we are talking about
in detail, or even see things in any real detail. We can write down
surprisingly strong principles that codify suprisingly superficial
mathematical ideas. Superficial in the sense of very blurry details.

When the details get more and more blurry, the power of the principles
emanating from the superficial mathematical ideas wane. They grow weaker
and weaker as things get more blurry. There is a proliferation of
incompleteness as we get more blurry. To compensate for this proliferation
of incompleteness, we want to make up stories. These stories get more and
more complicated and more and more far fetched, bringing more and more
features into the mix that are more and more blurry.

Harvey Friedman
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