Invariants, Boolean algebras and ACA

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 | References | Similar Articles | Additional Information

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. 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. This system corresponds to the assertion that (the arithmetic jump of ) exists for every set . 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 .

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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