[FOM] 808: Big Foundational Issues/2

[Harvey Friedman]

This continues https://cs.nyu.edu/pipermail/fom/2018-April/020907.html

1. IN PRECISELY WHAT SENSE DO WE EVEN HAVE A FOUNDATION FOR
MATHEMATICS? I.e., WHAT DOES ZFC REALLY ACCOMPLISH?

There is a well established mathematical style of definition, theorem,
proof that is readily recognized and readily accepted and readily used
by mathematicians across all fields and levels of experience. Let us
conveniently call this set theoretic foundations by the name ZFC,
which is the standard formal system associated with it. (But as we
will ultimately discuss, there are some important fragments and
variants).

ZFC may not be the preferred way of doing things for many
mathematicians. E.g., to varying extents, some may prefer rather
abstract formulations based on category theoretic ideas. But these are
considered very illuminating, somewhat illuminating, somewhat of a
frill, somewhat of a nuisance, a major nuisance, an incoherent
nuisance, and so forth, depending on the individual mathematician and
sometimes area of expertise. And, in fact, there is a standard way to
more or less capture advantages of such abstract formulations and
approaches by up front definitions in terms of set theory (e.g., the
approach taken at the beginning of Mac Lane, Categories for a Working
Mathematician).

NEVERTHELESS, there is a commonality to the ZFC approach, that every
competent mathematician is acceptably comfortable viewing themselves
as working within its framework. Of course, many mathematicians are
totally agnostic and unreflective about what "framework" they are
working in, and either don't care, or think that mathematics itself is
"beyond needing or resting on any kind of framework".

In contrast, for example, some people are seriously proposing some
sort of rather abstract rethinking of the notion of equality. At this
particular stage of research on this, the vast majority of
mathematicians want to completely ignore this, and when confronted too
aggressively with this, will recoil, viewing this as an unwanted
incoherent nuisance of a particularly obnoxious character, to be
avoided at all possible costs. The general disgust with things like
this is far greater than the general displeasure with sets.

Of course, in this series of postings, I am aiming at organizing my
thoughts and my results and future results in a way that adds new
insights. So what, for example, have I or do I want to say about all
this?

Quite a bit, and hopefully quite a bit that I have not previously said.

One thing I have worked on, announced in various talks, discussed on
the FOM, but did not write up proofs, is that, in a certain hard nosed
and perhaps surprising sense,

ZFC RESULTS FROM TAKING ALL TRUE STATEMENTS ABOUT THE HEREDITARILY
FINITE SETS OF A CERTAIN NATURAL FORM AND ADDING THE AXIOM OF
INFINITY.

I.e., ZFC naturally arises from going from the finite to the infinite.

This result(s) go back decades, but I want to revisit this with a
fresh (older) mind.

Of course, this addresses just one of many important aspects of issue
1. But it does gives us the following scenario.

Pretty much all mathematicians can get on the same page with regard to
the hereditarily finite sets. We can start with the semi formal
intuitively clear picture:

There are no sets of rank <= 0.
The sets of rank <= n+1 are the sets all of whose elements are sets of
rank <= n.

Even here there are some things that need to be said especially for
the neophyte. Specifically, here sets are extensional, and also there
are canonical terms for sets (this requires a little thought).

So there is a pretty clear picture of a structure (HF,epsilon), the
universe of hereditarily finite sets under the epsilon relation, and a
lot of cute little easy theorems can be profitably proved about this
structure and its uniqueness up to isomorphism, so that one has a kind
of satisfying mathematical theory going.

We are so far on very solid grounds, with a completely clear and
completely appropriate equality relation (avoiding any abstract
fiddling), and also a completely clear and completely appropriate
epsilon relation.

Of course, one can raise the issue: which is more fundamental, the
nonnegative integers or the hereditarily finite sets?

That is a tricky question, and the obvious answer is that each is more
fundamental in some important respects than the other. On a practical
note, there is the idea of my redoing what I have done bringing in the
nonnegative integers along with or instead of or whatever, the HF. But
I don't want to go there right now.

NOW HERE is the main point. There are obviously plenty of sentences
true (and very provable) about (HF,epsilon) that are false (and very
refutable) about any of the usual extensions of (HF,epsilon) such as
(V,epsion), whatever exactly V means.

BUT we teach all kinds of beautiful sentences about (HF,epsilon). and
these generally lift to (V,epsilon).

WHAT are these beautiful sentences? Can we algorithmically recognize
them? Can we axiomatize them?

So we have now pretty much caved out a little bit of a tree structure:

1. IN PRECISELY WHAT SENSE DO WE EVEN HAVE A FOUNDATION FOR
MATHEMATICS? I.e., WHAT DOES ZFC REALLY ACCOMPLISH?
1.1. FROM FINITE TO INFINITE.

************************************************************************
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 808th 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

Harvey Friedman