1
METAMATHEMATICS OF ULM THEORY
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
http://www.math.ohiostate.edu/~friedman/
June 24, 1999
November 25, 2001
INCOMPLETE DRAFT
Abstract. The classical Ulm theory provides a complete set of
invariants for countable abelian pgroups, and hence also for
countable torsion abelian groups. These invariants involve
countable ordinals. One can read off many simple structural
properties of such groups directly from the Ulm theory. We
carry out a reverse mathematics analysis of several such
properties. In many cases, we reverse to ATR
0
, thereby
demonstrating a kind of necessary use of Ulm theory.
1. INTRODUCTION.
Ulm theory is crucial in the theory of torsion abelian groups
(every element has finite order). It involves transfinite
recursion. Kaplansky, Infinite Abelian Groups, discussed two
test problems for Ulm theory:
a) Let G,H be countable torsion abelian groups, where G+G and
H+H are isomorphic. Then G,H are isomorphic.
b) Let G,H be countable torsion abelian groups, where G is a
direct summand of H and H is a direct summand of G. Then G,H
are isomorphic.
Kaplansky emphasizes that his test problems a,b immediately
follow from Ulm theory, and hence ATR
0
.
CONJECTURES: Test problems a) and b) are equivalent to
1
1
CA
0
over RCA
0
. Test problems a) and b) for countable reduced
torsion abelian groups are equivalent to ATR
0
over RCA
0
.
We formulate a number of other attractive test problems and
analyze them from the reverse mathematics point of view. One
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of our equivalents with ATR
0
over ACA
0
was mentioned in
[Si99], page 203.
We will mostly use ACA
0
as the base theory for the reversals.
This is very convenient. We do expect that in each case, with
additional work, one can replace ACA
0
with the weaker base
theory RCA
0
by deriving ACA
0
from RCA
0
plus the statement in
question. This should be worked out in the context of a
complete systematic metamathematical analysis of infinite
abelian group theory as represented, say, by [Fu70/73],
[Gr70], and [Ka69].
The main purpose of this paper is to show that the fairly
exotic logical nature of Ulm theory is required in the theory
of infinite abelian groups, in the sense that basic
structural facts require such logically exotic arguments.
However, the preponderance of results in countable infinite
Abelian group theory can be proved in such systems as ACA
0
and
RCA
0
. It would be interesting to have a full understanding of
countably infinite abelian group theory at these more typical
lower levels of reverse mathematics.
Countably infinite abelian group theory is a beautiful
context in which to do a systematic metamathematical analysis
via reverse mathematics because
a) Theorems in countably infinite abelian group theory are
naturally stated in the language of reverse mathematics,
without coding issues.
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 Fall '08
 JOSHUA
 Math, Order theory, Abelian group, Ulm, direct summand

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