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Transactions of the American Mathematical Society

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Invariants, Boolean algebras and ACA$_{0}^{+}$


Author: Richard A. Shore
Journal: Trans. Amer. Math. Soc. 358 (2006), 989-1014
MSC (2000): Primary 03B25, 03B30, 03C57, 03D28, 03D35, 03D45, 03F35, 06E05
DOI: https://doi.org/10.1090/S0002-9947-05-03802-X
Published electronically: April 13, 2005
MathSciNet review: 2187642
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Abstract: The sentences asserting the existence of invariants for mathematical structures are usually third order ones. We develop a general approach to analyzing the strength of such statements in second order arithmetic in the spirit of reverse mathematics. We discuss a number of simple examples that are equivalent to ACA$_{0}$. Our major results are that the existence of elementary equivalence invariants for Boolean algebras and isomorphism invariants for dense Boolean algebras are both of the same strength as ACA$_{0}^{+}$. This system corresponds to the assertion that $X^{(\omega)}$(the arithmetic jump of $X$) exists for every set $X$. These are essentially the first theorems known to be of this proof theoretic strength. The proof begins with an analogous result about these invariants on recursive (dense) Boolean algebras coding $0^{(\omega)}$.


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Additional Information

Richard A. Shore
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: shore@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03802-X
Received by editor(s): March 22, 2004
Published electronically: April 13, 2005
Additional Notes: The author was partially supported by NSF Grant DMS-0100035.
Article copyright: © Copyright 2005 Richard A. Shore

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