[FOM] Refuting the CH? (and a brief for aleph_2)

[Harvey Friedman]

On Wed, Jul 6, 2016 at 11:34 AM, Tom Dunion <tom.dunion at gmail.com> wrote:
> On July 5, 2016, Harvey Friedman said
>
> I suggest there may be grounds to be cautiously optimistic, and here's why.
>
Cautiously optimistic about what exactly?

> Beyond reasonable doubt, a resolution of the CH (in whatever form or
> fashion) has been a (the ?) holy grail of set theory from the beginning.

Resolution of the CH is normally thought of in the cont4ext of the
"matter of fact absolute truth" interpretation of set theory, based on
a Platonistic conception. Or at least in the context of first order
statements about (V(omega+2),epsilon). Prima facie, this point of view
seems dubious as there is so little information about what sets of
like just from the usual informal description, even for just
V(omega+2). This Platonism however bounces back some by saying that
there should be a clarification of what sets really are, and then
maybe we can move forward beyond the obvious principles, principles
known to leave masses of set questions open.

But furthermore there is essentially nothing in the entire
intellectual landscape, outside of concrete mathematics, where people
are even comfortable with "matter of fact absolute truth". Even in the
realm of physics, "matter of fact absolute truth" ran into a major
unexpected difficulty called quantum physics, and how one reconciles
that with matter of fact absolute truth is not clear.

Much more attractive is to step away from matter of fact absolute
truth to a more nuanced position. Namely, the idea that there are
"good" and "not so good" ways of doing set theory not resolved by the
usual axioms. And there is the conjecture that there is a "best way"
to do this outer set theory. And we should proceed to vigorously
uncover this "best way".

What is needed then is some criteria for comparing the merits of
different ways of doing outer set theory.

Now of course, in a sense, this (I think dubious) matter of fact
absolute truth approach to CH and values of |R| really kind of morphs
into such a nuanced position - while many practitioners still cling to
matter of fact absolute truth. Whereas they like to say things like
"this fits together in such and such (generally technical) way, and
this constitutes evidence for its absolute truth", it is more natural
to rethink this as saying that one has a "good" or "best" way to do
outer set theory in the (V(omega+2),epsilon) realm. BUT when making
the suggested move here, one need to ANALYZE and DEVELOP just what
general criteria one is using to justify that it is "good" or "best".

What Consistent Truth does here is to provide a really tangible
criteria for "good" or "best" way to do outer set theory. Namely, how
well are you conforming to the grammar of the vast bulk of fundamental
theorems in ordinary mathematics, or at least in real analysis?
Clearly, this criteria is very different from matter of fact absolute
truth. HOWEVER, it is a very transparent and easily understood
criteria that has much robustness, and clearly promises to generate a
huge new deep area of research which will easily absorb what might
naively be regarded as its fatal flaws - i.e., survive Inconsistent
Truth.

> A big part of the "grailness" is the long-standing irresolution of Cantor's
> claim.  But if we suppose (for the sake of argument) as a bit of
> counterfactual history, that between the world wars someone had advanced an
> argument against CH which met with near unanimous assent -- much as Choice
> eventually did, or like large cardinals today -- then the new grail likely
> would have been in determining the value of 2^{aleph_0}, not simply
> affirming or refuting CH.

I tried to launch |R| > omega_omega with Consistent Truth using

$[k]. For all f:R into R, there exist k reals, none of which is f at
the sum of the others.

and found that this, too, is equivalent to just notCH. I have found

$[k]'. For all f:R into R, there exist k reals drawn from any perfect
set, none of which is f at the sum of the others.

and see that this is equivalent to |R| > omega_omega. But I didn't
post on this because I am still looking for something yet more
fundamental. I can also avoid the perfect sets and do this if I go to
functions of several variables - something I do not want to do yet. I
want to try to keep everything really extremely simple until I am
forced (no pun intended) to add new ingredients.
>
> The independence results of Godel/Cohen might then have been primarily
> understood as telling us about the limitations of ZFC, not about the
> inherent vagueness of the CH, or the nebulousness of the Reals.  Even
> Lusin's claim in the 1920s, re the cardinality of co-analytic sets ("We
> don't know, and we shall never know") might have been seen in a similar
> light, that is, we need additional axioms beyond ZFC.

I posted earlier on the FOM, a Consistent Truth approach to getting
PD, which shows that the cardinality of co-analytic sets is countable
or c. Combining the two, you get that uncountable co-analytic sets
have cardinality at least omega_2. I haven't returned to the PD matter
yet, as I have my hands full with the CH.
>
> There are now many trails leading to aleph_2; Martin's Maximum, Proper
> Forcing Axiom, Open Coloring Axiom(s), other work of J.T. Moore and
> collaborators... There looks to be a rich and coherent theory with this
> value for the continuum, respecting many earlier results and concepts,
> whether strong Fubini theorems, cardinal characteristics, or random real
> forcing.  Interestingly, there don't seem to be principles or rules of thumb
> of the form "Here's why aleph_2 is likely ruled out" or "If you can get
> aleph_2, a little tweak always seems to get you to aleph_k"

So the challenge is to avoid appeals to matter of fact absolute truth
- which I think will consistently fail - and come up with the relevant
general interest transparent criterion for "good" and "best" ways to
do outer set theory,
>
> Let me close with a remark intended to be provocative (in a good, not
> antagonistic sense) -- Eighteenth century scientist Joseph Priestly isolated
> something he considered "de-phlogisticated" air; it took Lavosier to
> recognize there was actually a newly isolated element, oxygen!   I wonder if
> we are not looking at more and more evidence, at a level most scientists or
> detectives would find compelling.

I never liked the alleged "link" between outer set theory and physical
science. There is a whole complex web of interacting experiments with
unimpeachable conclusions/observations, and it is hard to make a case
that one has such in the case of outer set theory. FORTUNATELY, we do
not need to adopt matter of fact absolute truth in order to make
principled progress.

Harvey Friedman