[FOM] 811: Big Foundational Issues/3

[Harvey Friedman]

I am in the middle of finishing a fairly long manuscript with proofs
backing up versions of my previous
https://cs.nyu.edu/pipermail/fom/2018-May/020954.html

Considerable reworking of exact formulations have taken place, but the
basic points and basic thrusts are fully in tact - at least as of now.

IN THE MEANTIME, Heck wrote
https://cs.nyu.edu/pipermail/fom/2018-May/021001.html

I surmise that neither Shipman nor I (certainly not I) are referring
to any notion of "ZF(C) on firm foundation" as in any way referring to
"justification that it is consistent", or "justification that it is
true".

Rather: in the sense that it is robustly special in the realm of
theories of sets.

PHENOMENON: ZFC has stood the test of time by the mathematical
community as the generally preferred foundation for mathematics, not
anything weaker and not anything stronger. At least so far. And even
if ZFC gets overhauled with more axioms, say large cardinal axioms,
likely largely (but not entirely) because they get enough thrilling
uses (either for existing projects or new sufficiently thrilling
projects), there remains the question of just why the mathematical
community fixed on ZFC almost without dissent, for so long - ALMOST
ONE HUNDRED YEARS - until such applications of large cardinals appear
(whenever the critical mass actually does appear)..

A RIVAL: Z(C). See below.

ROLE OF MODERN f.o.m.: EXPLAIN ThIS PHENOMENON. EXPLAIN THIS
PHENOMENON AS BEST WE CAN.

BEST EXPLANATION TO DATE, WHICH IS A REASONABLY CREDIBLE EXPLANATION:
The work I am referring to of mine, and I am trying to finish a major
paper with proofs right now on this. Also, some of my results apply to
Z(C) with perfectly reasonable modifications. So Z(C) also gets picked
out robustly.

CAN WE FIND A BETTER EXPLANATION? Yes, but I don't have one right now.

DID I CHANGE THE RULES OF ENGAGEMENT HERE? In a sense yes. The OLD
RULES OF ENGAGEMENT go something like this: show me exactly why you
believe in ZF(C). The fact that things are (obviously) true in the
finite doesn't mean that they are true in the infinite, or even free
of contradiction in the infinite. In fact, you can argue perfectly
well that they have absolutely nothing to do with each other.

*THE OLD RULES OF ENGAGEMENT THROUGHOUT SERIOUS NEW f.o.m. ARE
HACKNEYED AND OUTDATED AND LEAD DOWN POINTLESS RABBIT HOLES. Hilary
Putnam actually made a special point of this. (actually I don't think
he went far enough, being too supportive of at least some pointless
rabbit holes). THE NEW RULES ALLOW US TO MOVE FORWARD WITH UNEXPECTED
DEPTH AND VIGOR*

This situation is becoming very typical in modern f.o.m. For EXAMPLE:

OLD RULES OF ENGAGEMENT ON CH: Argue that it is true or false based on
"set theoretic evidence".

NEW RULES OF ENGAGEMENT ON CH: I won't take up space here, but see my
"Principles of Consistent Truth" on my website, with totally new rules
of engagement. Of the current philosophically oriented set theorists,
the one I think that comes closest to being friendly to these totally
new rules of engagement is Donald A. Martin. This is yet another case
where the old rules of engagement lead down pointless rabbit holes.

CAN I INTERPRET MY ENTIRE f.o.m. EFFORT AS ONE OF ESSENTIALLY
ABANDONING OLD RULES OF ENGAGEMENT THAT I PERCEIVE LEAD DOWN POINTLESS
RABBIT HOLES, AND REPLACING THEM BY NEW RULES OF ENGAGEMENT INSTILLING
NEW DEPTH AND VIGOR? Maybe, and I will look into this systematically.

************************************************************************
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 811th 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 #1-799 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

800: Beyond Perfectly Natural/6  4/3/18  8:37PM
801: Big Foundational Issues/1  4/4/18  12:15AM
802: Systematic f.o.m./1  4/4/18  1:06AM
803: Perfectly Natural/7  4/11/18  1:02AM
804: Beyond Perfectly Natural/8  4/12/18  11:23PM
805: Beyond Perfectly Natural/9  4/20/18  10:47PM
806: Beyond Perfectly Natural/10  4/22/18  9:06PM
807: Beyond Perfectly Natural/11  4/29/18  9:19PM
808: Big Foundational Issues/2  5/1/18  12:24AM
809: Goedel's Second Reworked/1  5/20/18  3:47PM
810: Goedel's Second Reworked/2

Harvey Friedman