[FOM] 550: Foundational Methodology 1/Maximality

[Harvey Friedman]

Firstly, I have posted a detailed extended abstract:

86. Conservative Growth: A Unified Approach to Logical Strength,
October 12, 2014, 15 pages
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/

This supersedes my FOM postings #547 and #549.

Now I come to foundational methodology, and maximality in set theory.

I have been participating in a small specialized email list of about
70. The principal contributors there have been unwilling to move over
to the FOM for incoherent reasons (I think they have more coherent
reasons but are not willing to reveal them). Sometimes I write some
major pieces for it, and I have been posting the major content from
them here on the FOM.

I am now getting into making more serious postings there, and want to
make sure that I get the more major content presented here.

There has been some persistent discussion there concerning "maximality
in set theory". Yet there are diverse points of view concerning what
this means and whether it really makes sense and what it entails.
There is a folklore idea, that has definitely had some substantial
following over the years, that "maximality in set theory" is
sufficiently clear to generate the axioms of ZFC and more. However,
even that has been seriously questioned - at least openly on this
small email list. In particular, it has been seriously questioned
whether or not "maximality" is sufficiently clear to generate AxC.

So I have gotten interested in

1. Going deeply into what "maximality in set theory" really means, or can mean.
2. Whether there is a fundamental idea of "maximality in set theory"
that generates AxC. Same for other axioms of ZFC.
3. Whether there is a nontrivial fundamental idea of "maximality in
set theory" at all that meets certain fundamental criteria.

This is a good illustration of my modus operandi. Use philosophy and
philosophical considerations to generate foundational programs. Refine
the foundational programs according to complaints made by
philosophers. And so forth. This is what I call Ping Pong. You get a
series of foundational programs - many are refinements of previously
presented ones - that are of successively more and more interesting to
philosophers and to more and more mathematicians - particularly
mathematicians with philosophical interests.

This emphasizes the social side. For the foundationalist, the
foundational programs so generated are what is of fundamental
intrinsic importance, How I arrived at them - be it a complex
interactive process with philosophers and others - is secondary. The
fact that what results is immediately understandable to "everybody"
and generates delicious mathematical complexities of a very appetizing
nature is also secondary, but highly indicative and highly correlated
with the legitimacy and power of the foundational programs.

This particular example of genuine foundationalism in action is a very
good one. Specifically,

By "maximality in set theory" I mean more specifically the idea that
"the set theoretic universe is as large as possible". This is meant in
an intrinsic sense - a metaphysical sense, and possibly in an
epistemological sense. I will call this "intrinsic maximality".

There is another kind of "maximality in set theory" that has emerged,
which is meant in an extrinsic sense. Something like "mathematics is
best done without making any restrictions on the set theoretic
universe". I will call this "extrinsic maximality".

Apparently these days "extrinsic justifications" are more popular than
"intrinsic justifications". I was a bit surprised to find that out.
This could be generational.

As a foundationalist, here is my take on this situation. It has a lot
in common with my take on just about any situation in foundations of
mathematics.

A. Every account I have seen or heard about in favor and against
intrinsic and extrinsic maximality has some serious merits and some
serious demerits. Both in terms of coherence and in terms of
persuasion.
B. I don't have the slightest idea how coherent or how persuasive
formulations of intrinsic and extrinsic maximality will be or can be.
C. I have no preconceived agenda in the sense of even tending to
adhere to a position on intrinsic and extrinsic maximality - and would
not have one even if I succeeded in obtaining greatly clarified and
illuminating presentations of forms of them.

In fact, from this strongly foundationalist perspective, EVERYTHING in
f.o.m. is UP FOR GRABS, and can greatly benefit from full blown
reconsideration. EVERYTHING nontrivial that we have ever said in
f.o.m. that is reasonably convincing can likely be made MASSIVELY MORE
convincing, and likely can be subjected to novel DEVASTATING ATTACKS,
which by the way, can be STRONGLY DEFENDED AGAINST, etcetera/

Thus it is always CRUCIALLY IMPORTANT to reconsider EVERYTHING.

However, this does not mean that I should be arbitrary in just what I
am reconsidering. SOME reconsiderations do not look rewarding or
promising - do not look like they will lead anywhere. OTHER
reconsiderations may appear to be highly promising. With a finite
amount of time available, one needs to be careful to make strategic
choices.

There is a kind of UNIVERSAL MOVE that is of great importance in this
FOUNDATIONAL METHODOLOGY.

a. We are often searching for a coherent formulation of a notion or
principle. We struggle. The existing attempts are deeply flawed. We
can't improve on them, or maybe we can improve on them only
incrementally. There seems to be a barrier to something fully
coherent.

b. We make the MOVE. We switch to formulating general criteria for a
coherent formulation. We then prove that these general criteria cannot
be met.

This is a crucially important kind of MOVE which can turn a failed
foundational program into a successful - perhaps even spectacularly
successful - one.

In the next posting, I will continue, and we shall see what has come
out of my failed attempts thus far to deal with intrinsic maximality.

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

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 550th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM

Harvey Friedman