MetaUlmThy112501

MetaUlmThy112501 - 1 METAMATHEMATICS OF ULM THEORY by...

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1 METAMATHEMATICS OF ULM THEORY by Harvey M. Friedman Department of Mathematics Ohio State University http://www.math.ohio-state.edu/~friedman/ June 24, 1999 November 25, 2001 INCOMPLETE DRAFT Abstract. The classical Ulm theory provides a complete set of invariants for countable abelian p-groups, 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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2 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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MetaUlmThy112501 - 1 METAMATHEMATICS OF ULM THEORY by...

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