[FOM] Continuum/Letters/Clarification

[Harvey Friedman]

My recent postings Feferman/Continuum, and MathDept/Maddy are copies of
letters to Feferman and Maddy from another setting.

Since these letters contain no material from Sol or Pen, and stand alone, I
felt that I should put them on the FOM as my self contained statements.

But in my haste, I forgot to explain this when I made these two postings,
as I was just focused on the stand alone totally self contained nature of
these two postings. It hadn't occurred to me at the time that the FOM
wouldn't know the context, and also that the FOM cannot necessarily expect
to see any responses from Feferman or Maddy. I will ask them if they will
approve of me putting up their responses on the FOM.

I would like to continue to post letters to colleagues up letters to
scholars provided they are entirely self contained and do not contain any
material from them. Otherwise, I will be asking them for permission.

In the self contained Feferman/Continuum, I use capital letters to
distinguish my text from Sol's writings. Here is a copy without capitals.
Sol's original paper:  http://math.stanford.edu/~feferman/papers.html, #85.

*************************************************

I wonder if these processes are the products of evolution, peculiar to our
brains, or whether they are more fundamental. If brain related, can
something be said about the brain mechanisms involved, and the relevant
brain organization? If more fundamental, then can we replace our existing
foundational schemes, which are very powerful and robust and arguably
convincing, with new more fundamental foundational schemes? Obviously,
there may be a combination of the evolutionary and the fundamental, a
messier situation to deal with.

Our foundational schemes have been careful to remove any reference to such
materials or objects. What is to be gained by carefully putting them back
in? The crudest form of putting them back in is to simply add inert
urelements to set theory. Some well known interesting things happen when
you do this.

Again, what is to be gained by dealing more directly with these ideal world
pictures? In my
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#64,
 i consider this but only with the deliberate goal of getting very strong
interpretation power.

The lesson of
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#64
is simply that extremely little is needed to go a very very very long way
in terms of interpretation power. Pi01 statements come from the
interpretation power. Thus if a strong set theory t is interpreted in what
is arguably inherent in a simple mental picture, (picture accompanied with
a little text), then this constitutes a proof of any pi01 consequence of
the strong set theory.

Note that you cannot use just the soundness of a picture (with a little
text) to prove even pi02 statements this way - at least not without making
the picture much more involved and problematic.

"completely clear conceptions". This is an enormously important notion, and
it is to me highly deserving of great attention.

You think that the ring of integers is a completely clear conception. So
you think that truth in full is applicable to the ring of integers. You
think that (n,pow(n),epsilon) is not a completely clear conception, and in
fact far from a clear conception. And that it is not the case that truth in
full is applicable. I believe that as a consequence, your belief applies
equally well to the ring of real numbers with a distinguished predicate for
the integers, (r,z,+,dot). I certainly see a greater level of clarity about
the ring of integers than about the ring of reals with the integers. But i
also see a greater level of clarity about the integers of magnitude <= 2^
2^100 under partial addition and multiplication, than i see about the ring
of integers. In fact, i also see a greater level of clarity about the
integers of magnitude <=  2^100 under partial addition and multiplication,
than i see about the integers of magnitude <= 2^2^100 under partial
addition and multiplication. A greater level of clarity about the integers
of magnitude <= 100 under partial addition and multiplication than i see
about the integers of magnitude <= 2^100 under partial addition and
multiplication.

It is my impression that you are hopeful that an arbitrary first order
statement about a "completely clear conception" - as a relational structure
- can, with hard work and hard reflection and time be proved or refuted.

Of course there is the problem that the property in question may be
incomprehensible for various reasons, including being too long to state, or
too convoluted.

So let's say for the sake of argument that the property in question is
stated in a form that is completely "normal" for typical mathematics being
worked on by professionals.

Then what optimism do you have? The statement in question, even if typical
mathematically, may be known to be equivalent to the consistency of some
extremely strong set theory.

Thus there is the real prospect of us never being able to prove or refute
interesting statements in the ring of integers? Does that prospect, and the
way it arises, cause you to rethink your feelings about "completely clear
conceptions"?

I believe that the same kind of profound natural incompleteness is already
present in the integers of magnitude at most 2^100. That even this context
is inextricably linked up with the present large large cardinals.
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