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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Ranked structures and arithmetic transfinite recursion

Authors: Noam Greenberg and Antonio Montalbán
Journal: Trans. Amer. Math. Soc. 360 (2008), 1265-1307
MSC (2000): Primary 03F35, 03D45; Secondary 03C57, 03B30
Published electronically: October 23, 2007
MathSciNet review: 2357696
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Abstract: $ \mathsf{ATR}_0$ is the natural subsystem of second-order arithmetic in which one can develop a decent theory of ordinals. We investigate classes of structures which are in a sense the ``well-founded part" of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian $ p$-groups, and countable, compact topological spaces. Using computable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to $ \mathsf{ATR}_0$. The reductions themselves are also objects of interest.

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Additional Information

Noam Greenberg
Affiliation: Department of Mathematics, Notre Dame University, Notre Dame, Indiana 46556
Address at time of publication: School of Mathematics, Statistics and Computer Science, Victoria University, Wellington, New Zealand

Antonio Montalbán
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Received by editor(s): August 29, 2005
Published electronically: October 23, 2007
Additional Notes: We would like to thank our advisor Richard A. Shore for introducing us to questions that are discussed in this paper and for many useful conversations. Both authors were partially supported by NSF Grant DMS-0100035. This paper is part of the second author’s doctoral thesis.
Article copyright: © Copyright 2007 American Mathematical Society