FOM: 49:Ulm Theory/Reverse Math

[Harvey Friedman]

This is the 49th in a series of self contained postings to fom covering a
wide range of topics in f.o.m. Previous ones are:

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.
2:Axioms  11/6/97.
3:Simplicity  11/14/97 10:10AM.
4:Simplicity  11/14/97  4:25PM
5:Constructions  11/15/97  5:24PM
6:Undefinability/Nonstandard Models   11/16/97  12:04AM
7.Undefinability/Nonstandard Models   11/17/97  12:31AM
8.Schemes 11/17/97    12:30AM
9:Nonstandard Arithmetic 11/18/97  11:53AM
10:Pathology   12/8/97   12:37AM
11:F.O.M. & Math Logic  12/14/97 5:47AM
12:Finite trees/large cardinals  3/11/98  11:36AM
13:Min recursion/Provably recursive functions  3/20/98  4:45AM
14:New characterizations of the provable ordinals  4/8/98  2:09AM
14':Errata  4/8/98  9:48AM
15:Structural Independence results and provable ordinals  4/16/98
10:53PM
16:Logical Equations, etc.  4/17/98  1:25PM
16':Errata  4/28/98  10:28AM
17:Very Strong Borel statements  4/26/98  8:06PM
18:Binary Functions and Large Cardinals  4/30/98  12:03PM
19:Long Sequences  7/31/98  9:42AM
20:Proof Theoretic Degrees  8/2/98  9:37PM
21:Long Sequences/Update  10/13/98  3:18AM
22:Finite Trees/Impredicativity  10/20/98  10:13AM
23:Q-Systems and Proof Theoretic Ordinals  11/6/98  3:01AM
24:Predicatively Unfeasible Integers  11/10/98  10:44PM
25:Long Walks  11/16/98  7:05AM
26:Optimized functions/Large Cardinals  1/13/99  12:53PM
27:Finite Trees/Impredicativity:Sketches  1/13/99  12:54PM
28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM
28':Restatement  1/28/99  5:49AM
29:Large Cardinals/where are we? I  2/22/99  6:11AM
30:Large Cardinals/where are we? II  2/23/99  6:15AM
31:First Free Sets/Large Cardinals  2/27/99  1:43AM
32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM
33:A Variant  3/4/99  1:52PM
34:Walks in N^k  3/7/99  1:43PM
35:Special AE Sentences  3/18/99  4:56AM
35':Restatement  3/21/99  2:20PM
36:Adjacent Ramsey Theory  3/23/99  1:00AM
37:Adjacent Ramsey Theory/more  5:45AM  3/25/99
38:Existential Properties of Numerical Functions  3/26/99  2:21PM
39:Large Cardinals/synthesis  4/7/99  11:43AM
40:Enormous Integers in Algebraic Geometry  5/17/99 11:07AM
41:Strong Philosophical Indiscernibles
42:Mythical Trees  5/25/99  5:11PM
43:More Enormous Integers/AlgGeom  5/25/99  6:00PM
44:Indiscernible Primes  5/27/99  12:53 PM
45:Result #1/Program A  7/14/99  11:07AM
46:Tamism  7/14/99  11:25AM
47:Subalgebras/Reverse Math  7/14/99  11:36AM
48:Continuous Embeddings/Reverse Mathematics  7/15/99  12:24PM

REVERSE MATHEMATICS OF ULM THEORY

There is a nearly completed manuscript which includes the following
results. It will eventually appear on my website.

Consider the following statements for countable Abelian groups.

1. Either G is embeddable into H* or H is embeddable into G*.
2. There is a direct summand K of G and H such that every direct summand of
G and H is embeddable into K.
3. There is a direct summand J of G* and H* such that every direct summand
of G* and H* is a direct summand of J.
4. In every infinite sequence of groups, one group is embeddable in a later
(different) group.
5. In every infinite decreasing chain of groups, one group is embeddable in
a later group.

Here G* is the direct sum of countably many copies of G. In 5, a decreasing
chain of groups is a sequence of groups G1,G2,..., where each Gi+1 is a
subgroup of Gi. Here we mean literal subgroup, not just up to isomorphism.

Reduced means no divisible subgroup. Torsion group means every element is
of finite order.

THEOREM 1. For reduced p-groups, each of 1-5 are provably equivalent to
ATR_0 over RCA_0. This is also true for any specific prime p. For reduced
torsion groups, each of 2,3 are provably equivalent to ATR_0 over RCA_0.
1,4,5 are false for reduced torsion groups.

THEOREM 2. For p-groups, each of 1-5 are provably equivalent to ATR_0 over
RCA_0. This is also true for any specific prime p. For torsion groups, each
of 2,3 are provably equivalent to ATR_0 over RCA_0. 1,4,5 are false for
torsion groups.

Theorem 2 may be somewhat surprising since obvious proofs lie in
pi-1-1-CA_0. Additional work is needed to stay within ATR_0.

The reversals of 4 and 5 rely on Richard Shore's:

On the strength of Fraisse's conjecture, in: Logical Methods, In Honor of
Anil Nerode's Sixtieth Birthday, Brikhauser, 1993, 782-813.

Actually, we need a small refinement of Shore for 5, which looks
straightforward.

I am indebted to Paul Eklof for valuable discussions, especially concerning
the paper:

Jon Barwise and Paul Eklof, Infinitary Properties of Abelian Torsion
Groups, Annals of Mathematical Logic, 1970, 25-68.

PS: The proof of 2,3 for countable Abelian p-groups in ATR_0 relies on a
technical generalization of Ulm's theorem which I sent out to be checked
for verification.