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*To*: fom@math.psu.edu*Subject*: FOM: 49:Ulm Theory/Reverse Math*From*: Harvey Friedman <friedman@math.ohio-state.edu>*Date*: Sat, 17 Jul 1999 15:21:26 +0100*Sender*: owner-fom@math.psu.edu

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.

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