[FOM] Book appearing: Formalization without Foundationalism: Model Theory and the Philosophy of Mathematical Practice

[Harvey Friedman]

On Mon, Jan 8, 2018 at 11:48 PM, John Baldwin <jbaldwin at uic.edu> wrote:
> I take this opportunity my about to be published book.
>
> Bound copies of the proofs of the above book will be available at the
> Cambridge Press booth at the annual Joint Mathematics Meetings in San Diego.
> The book will be available a bit later.
>
> Here is a lightly edited version of the book proposal.
>

John, thanks very much for putting out that substantive "edited form
of the book proposal".

I am briefly responding to what you wrote, and I expect to have a lot
more to say after the book is available.

A major part of the current ideology in applied model theory today is that

*) the so called Goedel phenomena is a wrong turn and totally
unnecessary turn in the history of mathematics, that needs to be, and
is fairly easily, separated from the "good" or "productive" part of
mathematics. This is done through "tameness", with one particularly
clear embodiment being realized by 0-minimality, something with its
origins in Grothendieck's "tame topology" and later firmed up and
investigated deeply by applied model theorists.

Obviously, applied model theorists did not have any incentive, nor did
they believe in, the matter of bridging the gap between Godel
phenomena and "real" mathematics. The assumption is simply that that
gap is very wide and totally necessary.

However, as the f.o.m. revolution of the 21st century continues, of
course at a not quick enough pace for many of us, the gap is being
removed.

Also, closely related, the idea that there are appropriate independent
autonomous foundations for separate parts of mathematics, where the
foundational schemes do not interact deeply, is also being refuted.

Of course, in this very early part of the 21st century, the known
results are just not yet strong enough to make all of this completely
obvious, as it likely will by the end of the 21st century, and without
any doubt in my mind, in a few more centuries. That is the way things
are clearly evolving.

SOME EXPECTED FUTURE DEVELOPMENTS TOWARDS FIRMLY EMBEDDING THE GOEDEL
PHENOMENA EVERYWHERE IN NORMAL MATHEMATICS

1. No less than Carl Ludwig Siegel (CLS) is famously created for
establishing a decision procedure for the solvability of quadratic
equations in the integers. (By the way, somebody questioned whether
this has been done for the rationals. References?)

2. We know that the solvability of systems of quadratic equations in
the integers is Goedelian. So CLS is banging his head potentially
against the Goedelian. POSSIBLE FUTURE: Very few simultaneous
quadratic equations in very few variables are already algorithmically
unsolvable.

3. At the most EXTREME that I know of, there is the possibility that
{n^2 + m^3: n,m in Z} is algorithmically unsolvable. If you find that
unimaginable, fool with that expression a tiny bit, and then see what
you think.

4. There is an integer 0 < r < 2^30 such that the solvability in
integers of n^2 + m^3 = r (or use your favorite alternative very very
simple expression, if you like, maybe with coeffieicnets also < 2^30 -
is independent of the ZFC axioms. Provable from large cardinals, but
not in ZFC.

5. It is possible to take, rather generally, simple combinatorial
properties of long well orderings (too long for ZFC), which are
sharper than well known such provable in the nonnegative integers,
and, in a simple strategic uniform way, restate them in the rational
numbers, so that the resulting statements can and can only be proved
using large cardinals. As of December 2016, this is more or less
essentially here.

6. 5 will be further pushed down into elementary number theoretic
relations between finite sets of integers that will gradually be
extremely simple and totally strategic. A probably significant but
small step is in my recent series on 1-dimensional incompleteness.

7. These Goedelian combinatorial matters will be slowly moved into the
realm of finitely presented groups. Then the longstanding connection
with geometry/topology will be firmed up for this, to put Goedel
inexorably into geometry/topology.

8. At a more abstract level, we already know that for consistent
single sentences in predicate calculus, there is no maximal such under
interpretability. But the natural proof, and in fact the only proof I
am aware of, passes through the heart of Goedel. OK, sure, you want to
call predicate calculus Goedelian and remove it as well (at least the
extreme form of Baldwin). But what about finite systems of equations,
with the tacit "there exists at least two objects" - same question
about maximal interpretation power. This also passes through Goedel.
And this can be made even more algebraic.

9. There will be an intelligible sentence in the usual (decidable)
ordered field of real numbers with the following properties.

i. It is true.
ii. Any proof of it in ZFC is too large to fit in the existing worldwide web.

To be continued, if people are interested!

Harvey Friedman



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