[FOM] Solving set theoretic problems?/2

[Harvey Friedman]

We discuss some issues arising with
http://www.cs.nyu.edu/pipermail/fom/2016-May/019806.html

There are now a few research programs aimed at choosing principled
methods for "solving" set theoretic problems. Here I want to try to
put them into a more coherent perspective.

For those with a hard nosed Platonist viewpoint, that set theoretic
statements, at least those about V(omega + 2), are matter of fact,
probably the kind of thing I am doing is not going to be considered
convincing or even particularly relevant. I think it is also clear
that for most but not all of such hard nosed, diehards, there is
probably no research program around that is convincing or particularly
relevant. For those few hard nosed diehards who do think that there is
a research program that is convincing or truly relevant, they are
probably going to face a seriously uphill battle to convince the hard
nosed diehards, generally.

But there are soft nosed realists who fall short of Platonism, but
still think that there is a pretty much uniquely "convincing" way of
doing set theory. These people are in search for some clear and
attractive guiding principles about how to solve set theoretic
problems (that are of course not solvable by the ZFC axioms). These
people may be particularly interested in looking at proposals for
principled ways of "solving" set theoretic problems.

>From http://www.cs.nyu.edu/pipermail/fom/2016-May/019806.html it is
clear that there are a number of interlocking research programs being
proposed, with some illustrative starting dvelopments. Some
clarifications are in order.

1. CONSISTENT TRUTH PROGRAM

In the consistent truth programs, we do not, at least not initially,
bring the Borel Universe into the picture.

Here we identify a language L of mathematical statements, and explore
the statement

CONSISTENT TRUTh FOR L. CT(L). Every sentence in the language L that
is consistent with ZC is true.

over ZFC. Now obviously this is completely ridiculous if the language
L is closed under negation. CT(L) either gives us nothing new or is
inconsistent with ZFC - i.e., worthless.

Also, for any language L, there is the language -L, consisting of the
negations of the sentences in L (maybe of course put in preen or nice
logical form to make it look more attractive). And then CT(-L) will
point us to the opposite conclusions from CT(L).

So it is clear that under this Consistent Truth Method, the crucial
conceptual matter is the choice of languages L.

>From looking at
http://www.cs.nyu.edu/pipermail/fom/2016-May/019806.html and the
references there to earlier FOM postings, where I "prove" not CH and
PD, I have chosen certain languages L. Actually, only the "proof" of
not CH falls into the Consistent Truth Program as we are discussing it
now.

As an initial step, I have determined the exact status of CT(L) for a
special simple fragment of the language of sentences

(for all f:R into R)(there exists x,y in R)(for all n in Z)(phi)

where phi is a propositional combination of formulas involving
f,x,y,n,+,= in the obvious way. The status is simply that for this
special simple fragment L, CT(L) is provably equivalent to not CH over
ZFC. See http://www.cs.nyu.edu/pipermail/fom/2016-May/019805.html

It is natural to round out this language to the much larger language
of sentences

(for all f:R^n into R^m)(there exists x_1,...,x_r in R)(for all
n_1,...,n_s in Z)(phi)

where phi is a propositional combination of formulas involving
f,x's,n's,+,< in the obvious way. And even further. The natural
languages of this rough kind to consider are rather considerable, and
it is really premature to go into this further at this early point.

Now at some point in the natural enlargement of such languages, say
brining in multiplication and/or additional quantifiers, we are going
to get enough expressive power that CT(L) will be inconsistent. E.g.,
things like "every real is constructible" will be expressible, which
contradicts not CH. And I am very interested in just what the
threshold here is - when, as we expand the languages L in this way, we
enter into the expressibility of "every real is constructible" and/or
some appropriate sort of universality. I made some initial attempts to
explore this matter in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019800.html

A point I emphasized in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019800.html is that even
if we run across an enlarged language L as above where we can express
"every real is constructible" or even have a strong kind of
universality, the actual instances involved are spectacularly
complicated not only on an absolute scale, but compared to normal
mathematical statements such as $,$' in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019805.html

This suggests a more nuanced program

2. SIMPLE CONSISTENT TRUTH PROGRAM

SIMPLE CONSISTENT TRUTH FOR L. SCT(L). Every sentence in the language
L that is reasonable simple (of simplicity standardly seen in normal
mathematics) and consistent with ZC, is true.

Thus in Simple Consistent Truth we are considering only languages L
with finitely many sentences.

The idea (conjecture) is that we can go very very far with languages L based on

(for all f:R^n into R^m)(there exists x_1,...,x_r in R)(for all
n_1,...,n_s in Z)(phi)

if we use SCT(L) instead of just CT(L).

3. CHOOSING LANGUAGES

We now want to address the Big Elephant in the Room. Why should we
choose L to be roughly

1) (for all f:R^n into R^m)(there exists x_1,...,x_r in R)(for all
n_1,...,n_s in Z)(phi)

instead of the dual

2) (there exists f:R^n into R^m)(for all x_1,...,x_r in R)(there
exists n_1,...,n_s in Z)(phi) ?

And then, consequently, be arguing for CH instead of not CH?

One approach to this issue is to argue that 1) is better than 2) on
the grounds that the dominate pull of actual mathematics - the
preponderance of interesting mathematical statements/theorems (among
these two) - is with 1) instead of 2).

We can actually go into the actual mathematical literature and see
what we find regarding this.

Of course, what is going to  happen is that we are not going to get
much actual data of the kind we are looking for here. What we are
likely to find is many cases that are reasonably analogous to the
1),2) dichotomy, where we see

3) (for all complicated/big things)(there exists simpler/smaller
things)(something yet simpler holds)

dominating over

4) (there exists complicated/big things)(for all simpler/smaller
things)(something yet simpler holds)

For a vivid example, consider all of those delicious rigorous calculus
theorems about continuous functions from R to R. Although the details
are all over the place, we always see that the interesting theorems
start with a universal quantifier over the complicated/big thing - the
continuous function - followed by existential quantifiers over the
simpler/smaller thing - the real numbers - followed by something even
simpler. Yes, in the front there is also universal quantifiers over
the simper thing (reals), but these can be nicely absorbed. The same
pattern holds for the theory of one complex variable, or several
complex variables. Also things like Taylor series, etcetera.

OR, ALTERNATIVELY, we could take a cue from math and expand our 1)
above directly to

1') (for all f:R^n into R^m)(for all x_1,...,x_r in R)(there exists
y_1,...,y_s in R)(for all n_1,...,n_t in Z)(phi)

which really doesn't buy us anything in general, but when we consider
tiny fragments as we have started to do with
http://www.cs.nyu.edu/pipermail/fom/2016-May/019805.html it may be
important.

So there seems to be little doubt that in some interesting senses, we
can take a cue from actual very important mathematics and choose 1)
over 2). This seems to be a phenomena that occurs throughout
mathematics. The most involved/largest objects are universally
quantified out in front.

This whole discussion of the Big Elephant in the Room does suggest
some rich enhancements of the Consistent Truth and Simple Consistent
Truth programs.

Specifically, start with an examination of the most classic
mathematical theorems of analysis - and maybe beyond analysis. Develop
languages L based on these classic theorems. Then analyze CT(L) and
SCT(L), where continuous or differentiable, or real analytic or
analytic functions are replaced by arbitrary functions. And maybe go
back and forth between the two situations, using each other to
motivate various L's. Of course, the truth values will be violently
changed. But we are really concerned with the mathematical and
intellectual naturalness of logical forms.

4. BOREL TRUTH PROGrAMS

There is another way to choose 1) over 2). Actually, it is a way of
choosing both 1) and 2), and making them compatible with each other!
What really happens is that under this Borel move, below, 2) seems to
yield nothing, whereas 1) yields a lot.

BOREL TRUTh FOR L. BT(L). Every sentence in the language L that is
true for Borel functions is true.

We had discussed previously an example where BCT(L) contradicts the
axiom of choice. The example is quite simple but needs some additional
or different primitives than what we have been discussing. See section
5 of http://www.cs.nyu.edu/pipermail/fom/2016-May/019791.html

With this caveat in mind, we can still look at BCT(L) for the line of
languages we have been discussing. Returning to the 1),2) discussed in
section 3 above, we expect, at least to an unknown extent to be
discovered later, that for 1), BT gives us exactly not CH, and for 2),
BT gives us nothing. I.e., for 2), BCT is provable in ZFC (even in
ZC). Also, we expect that the Borel statements with 1),2) are provable
or refutable in ATR_0. We already know everything in this paragraph
for the very basic 1),2) from
http://www.cs.nyu.edu/pipermail/fom/2016-May/019805.html

BOREL CONSISTENT TRUTH FOR L. BCT(L). Every sentence in the language L
that is true for Borel functions and consistent with ZC is true.

BCT is a way of avoiding the problem that BT violates the axiom of
choice for such simple L. We seem to have to go to something very
complicated to get into trouble with BCT(L). With BCT, we don't have
to decide between dual classes like 1),2). We can have both of them
simultaneously, or at least it seems this way, with one of them being
provable and the other having the real red meat. Naturally, of course,
the one where the outside quantifier universally quantifies over all
Borel functions.

5. BOREL/PD PROGRAM

BOREL/PD  FOR L. BPD(L). Every sentence in the language L that is true
for Borel functions is true for projective functions.

We have been focusing on not CH and have not developed this further
than what we have in
http://www.cs.nyu.edu/pipermail/fom/2016-April/019775.html

Here we are looking at languages with similar outside universal
quantifiers, but where the existential quantifier is over a richer
domain.

Harvey Friedman