FOM: F.O.M. = Reverse Math?

[Harvey Friedman]

Because of the frequent mention of reverse mathematics by Simpson, myself,
and several other subscribers in many contexts, I get the feeling that some
subscribers may be under the impression that I think that F.O.M. = Reverse
Math., or that I work exclusively in Reverse Math., or that I work mostly
in Reverse Math. This is not the case. Let me go through my numbered
postings, and indicate which ones are in reverse math with "YES" and the
ones that are not in reverse math with "NO." Also, as the "Central Issues
in Foundations" series unfolds, you will see that reverse math will be just
one out of a great many foundational topics discussed, several of which I
have contributed to. At the end of this posting, I will discuss some
features of Reverse Math that explains why I have so frequently discussed
this on the FOM.

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.  NO
2:Axioms  11/6/97.  NO
3:Simplicity  11/14/97 10:10AM.  NO
4:Simplicity  11/14/97  4:25PM  NO
5:Constructions  11/15/97  5:24PM  NO
6:Undefinability/Nonstandard Models   11/16/97  12:04AM  NO
7.Undefinability/Nonstandard Models   11/17/97  12:31AM  NO
8.Schemes 11/17/97    12:30AM  NO
9:Nonstandard Arithmetic 11/18/97  11:53AM  NO
10:Pathology   12/8/97   12:37AM  NO
11:F.O.M. & Math Logic  12/14/97 5:47AM  NO
12:Finite trees/large cardinals  3/11/98  11:36AM  NO
13:Min recursion/Provably recursive functions  3/20/98  4:45AM  NO
14:New characterizations of the provable ordinals  4/8/98  2:09AM  NO
14':Errata  4/8/98  9:48AM  NO
15:Structural Independence results and provable ordinals  4/16/98
10:53PM  NO
16:Logical Equations, etc.  4/17/98  1:25PM  NO
16':Errata  4/28/98  10:28AM  NO
17:Very Strong Borel statements  4/26/98  8:06PM  NO
18:Binary Functions and Large Cardinals  4/30/98  12:03PM  NO
19:Long Sequences  7/31/98  9:42AM  NO
20:Proof Theoretic Degrees  8/2/98  9:37PM  NO
21:Long Sequences/Update  10/13/98  3:18AM  NO
22:Finite Trees/Impredicativity  10/20/98  10:13AM  NO
23:Q-Systems and Proof Theoretic Ordinals  11/6/98  3:01AM  NO
24:Predicatively Unfeasible Integers  11/10/98  10:44PM  NO
25:Long Walks  11/16/98  7:05AM  NO
26:Optimized functions/Large Cardinals  1/13/99  12:53PM  NO
27:Finite Trees/Impredicativity:Sketches  1/13/99  12:54PM  NO
28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM  NO
28':Restatement  1/28/99  5:49AM  NO
29:Large Cardinals/where are we? I  2/22/99  6:11AM  NO
30:Large Cardinals/where are we? II  2/23/99  6:15AM  NO
31:First Free Sets/Large Cardinals  2/27/99  1:43AM  NO
32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM  NO
33:A Variant  3/4/99  1:52PM  NO
34:Walks in N^k  3/7/99  1:43PM  NO
35:Special AE Sentences  3/18/99  4:56AM  NO
35':Restatement  3/21/99  2:20PM  NO
36:Adjacent Ramsey Theory  3/23/99  1:00AM  NO
37:Adjacent Ramsey Theory/more  5:45AM  3/25/99  NO
38:Existential Properties of Numerical Functions  3/26/99  2:21PM  NO
39:Large Cardinals/synthesis  4/7/99  11:43AM  NO
40:Enormous Integers in Algebraic Geometry  5/17/99 11:07AM  NO
41:Strong Philosophical Indiscernibles  NO
42:Mythical Trees  5/25/99  5:11PM  NO
43:More Enormous Integers/AlgGeom  5/25/99  6:00PM  NO
44:Indiscernible Primes  5/27/99  12:53 PM  NO
45:Result #1/Program A  7/14/99  11:07AM  NO
46:Tamism  7/14/99  11:25AM  YES
47:Subalgebras/Reverse Math  7/14/99  11:36AM  YES
48:Continuous Embeddings/Reverse Mathematics  7/15/99  12:24PM  YES
49:Ulm Theory/Reverse Mathematics  7/17/99  3:21PM  YES
50:Enormous Integers/Number Theory  7/17/99  11:39PN  NO
51:Enormous Integers/Plane Geometry  7/18/99  3:16PM  NO
52:Cardinals and Cones  7/18/99  3:33PM  NO
53:Free Sets/Reverse Math  7/19/99  2:11PM  YES
54:Recursion Theory/Dynamics 7/22/99 9:28PM  NO
55:Term Rewriting/Proof Theory 8/27/99 3:00PM  NO
56:Consistency of Algebra/Geometry  NO


Here are some special features of Reverse Mathematics.

1. The problems range virtually continuously from easy to apparently
extremely difficult.

2. There is an extremely well defined foundational purpose of obvious
general intellectual interest. E.g., many years ago the interest was
completely obvious to some key members of the Harvard Math Dept in core
mathematics, who ran a seminar in it many years ago on their own
initiative. This is unusual for a topic in mathematical logic.

3. There is an underlying conjecture of obvious general intellectual
interest, which is quite striking and appears to be true. This is the
linear ordering of actual mathematical theorems under interpretability (or
consistency strength) over the base theory of present Reverse Mathematics.

4. The projected size and scope, even under the present Reverse
Mathematics, is absolutely enormous. For example, take the two volume set
on Abelian Group Theory by Fuchs, and restrict everything to countable
groups, where the essence of most everything there lives anyways. A very
significant percentage of the Propositions and Theorems in those books lend
themselves to significant Reverse Math treatment. This surprised me at
first. Many of these things are very difficult to handle. I would
conservatively estimate that a systematic treatment of just these two
volumes from just the present Reverse Mathematics viewpoint would yield at
least 10,000 pages of very interesting mathematical logic. In general, you
must have a far deeper knowledge of a Theorem amenable to Reverse Math than
you would simply to understand a proof of it. SAMPLE: See posting #49
above.

5. But it now seems clear that Reverse Mathematics is destined to have many
new forms which will incorporate an increasing amount of mathematical
contexts into its domain. I predict that a systematic development of the
Reverse Mathematics, once these new forms get established, would be
significantly larger than all of the other parts of pure mathematics
combined. Of course, it is absurd to think that support for an effort of
that magnitude would ever be available - or even should be made available.
But I expect that Reverse Mathematics is destined to interact deeply with
virtually all parts of pure and applied mathematics in a very satisfying
and profound way, with a virtually unlimited supply of satisfying problems
with substantial technical content.

6. It is hard to imagine any program in mathematical logic with such
properties. But why does Reverse Math stand out in this way? Because it is
currently our best and most comprehensive general formal approach to the
question "what is the logical structure of actual mathematics?" that we
have. And that question is such a central issue in the foundations of
mathematics.

7. Coming back to the title of this posting, of course the answer is NO.
But at the moment, Reverse Mathematics appears to be the most productive
*problem generator* in f.o.m.

8. Nothing said above about Reverse Mathematics is intended to minimize the
significant overlap that it has with an area called Recursive Mathematics.
Reverse Mathematics addresses "what is the logical structure of actual
mathematics?" whereas Recursive Mathematics addresses other aspects
(various forms of effectivity) of the structure of actual mathematics.