[FOM] the "best way to do set theory"

[Harvey Friedman]

On Thu, Jul 7, 2016 at 6:38 PM, Martin Davis <martin at eipye.com> wrote:
> In a recent FOM post, Harvey contrasted an "absolute truth" stance on set
> theory with what he sees as the more appropriate "good ways" and even a
> "best way" to do set theory when going beyond ZFC. To my mind putting the
> question in these terms is somewhat question begging.

The "matter of fact absolute truth" attitude to set theory is just one
of many equally defensible positions, no more compelling than others.

In fact, the extensions of ZFC that have been proposed, being proposed
now, or about to be proposed, in connection with open set theoretic
problems like CH,are far more compelling when thought of as "a better
way to do set theory" or "a best way to do set theory" than they are
as answering any call to matter of fact absolute truth.

The hope for "missing axioms of set theory" in the sense of
obvious-in-hindsight oversights has pretty much vanished. The
proposals that have been around or are being developed are far better
characterized as "proposed better ways to do set theory" or "proposed
best ways to do set theory" than they are as uncovering matter of fact
absolute truth. Whereas researchers engaged in such projects may think
of or justify their efforts as "uncovering matter of fact absolute
truth", it is clear that all but a few are going to instead think of
what they are proposing as "proposals for a better way to do set
theory" or "proposals for a best way to do set theory".

So what I am proposing is that we stop trying to think in terms of
"matter of fact absolute truth" and instead directly think of "better
and best ways to do set theory", which is an inevitable shift anyway.

And given this shift in motivation or perspective, we need to rethink
everything going back to basics at the most fundamental levels.

In uncovering "better or best ways of doing set theory", we have to
think in fundamental terms as to what convincing criteria should be
for judging "better" or "best" here. This sets the stage for the
Principle of Consistent Truth approach, with its criteria for the
choice of languages to be used for Principles of Consistent Truth.

> I have been a great fan of Harvey's work on Pi-0-1 statements that are
> provable only when the existence of a large cardinal is assumed. In many
> cases the statement in question is equivalent to the consistency of ZFC +
> "the existence of some large cardinal". To someone who is not a crude
> formalist or an ultra finitist, the truth or falsity of a Pi-0-1 statement
> is an objective fact (whether we will ever know which).

Historically and currently, there has been reasonably strong adherence
to matter of fact absolute truth for Pi01 sentences. In fact, far
stronger adherence for Pi01 sentences than for set theory even in just
the realm of statements in (V(omega + 3),epsilon) or even
(V(omega+2),epsilon), or second order logic formulations. However, the
denial of matter of fact absolute truth for Pi01 sentences is also a
perfectly defensible position, as defensible as other relevant
positions.

At this time, we do not have a clear enough idea of what we even mean
by "matter of fact absolute truth" to make really convincing decisions
as to the contexts in which "matter of fact absolute truth" applies.
Therefore the most appropriate and relevant way for a foundationalist
to operate without the constraints of any hard to defend positions on
such matters, but rather flexibly, responding to as many prima facie
coherent positions as is feasible.

> Why would Harvey
> even devote himself to finding nice combinatorial forms for such
> propositions, if he didn't believe they were likely mathematical truths of
> interest to mathematicians? Now there is Hilbert's famous dictum, that in
> mathematics for an entity to exist means no more or less than that the
> assumption of its existence is consistent.

As indicated above, I operate without accepting any dogma concerning
such issues. It is enough to be responsive to varied equally
defensible positions.

In fact, there is a very compelling way to come to the Concrete
Incompleteness program without getting close to adhering to any dogma.

There has been an official rule book for mathematical proof called ZFC
in place by around 1920 which has remained totally unchallenged for
competing mathematical purposes. The contexts in which ZFC is known to
be inadequate are minor and inessential from standard mathematical
viewpoints. In particular, they involve irrelevant excess generality
as measured by the original motivating mathematical concerns. So
leaving questions unresolved that rely on a kind of excess generality
that is not responsive to clear motivating concerns is considered
highly preferable to any kind of rethinking of the usual ZFC axioms
for mathematics.

So the plan for Mathematical Inompleteness is to entirely overthrow
this situation by showing that mathematics of an essential
fundamentally motivated mathematical character as judged by any normal
criteria used to judge important mathematical developments, can not be
developed within the ZFC framework. The revolutionary nature of this
plan does not in any way depend on adherence to any dogma.

Incidentally, given what I have just written here, one can wonder why
I have taken up "resolving CH" as my currently favored hobby (along
with matters surrounding piano performance). The answer is that even
though I have doubts as to the essential importance of actually
developing abstract set theory in its own right, in the realm of
general intellectual inquiry, the issue of how "good" or "best" it can
be done, and how this can be judged, is clearly interesting and
important in the context of general methodological issues that cut
across the entire intellectual landscape.

> Of course we know that we cannot hope for a mathematical proof of the
> consistency of large cardinal axioms. In fact given the increasing
> consistency strength as one marches up the large cardinal hierarchy, the ice
> on which one is marching gets ever thinner as one ascends. Kanamori's famous
> chart emphasizes this by placing "0=1" at the top! Certainly the illusion of
> certainty is gone, but we are no worse off than empirical scientists.

I'm not sure what "worse off" means here. But empirical scientists
have a big tool that we do not have here. Namely, they have
incontrovertible experimental apparatus, so they can test hypotheses
of varied kinds by experimentation, including confirmation of
universal statements. Incidentally, I got interested in whether there
can be anything like empirical verification of the consistency of ZFC
and large cardinals, and have written about how this might be achieved
through computer searches and my work on incompleteness. I need to
update this work taking into account more recent developments.
>
> I think it also worth noting that because Pi-0-1 propositions can be
> expressed as a polynomial equation having no natural number solutions, a
> counter-example to a false Pi-0-1  proposition can be verified by a finite
> number of additions and multiplications of integers.

Of course, this isn't sensitive to the nuances concerning what is
meant by "can be verified" here.

Harvey Friedman