Comparing theorems of hyperarithmetic analysis with the arithmetic Bolzano-Weierstrass theorem

Author:
Chris J. Conidis

Journal:
Trans. Amer. Math. Soc. **364** (2012), 4465-4494

MSC (2010):
Primary 03F35; Secondary 03D80

DOI:
https://doi.org/10.1090/S0002-9947-2012-05416-X

Published electronically:
April 18, 2012

MathSciNet review:
2922598

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1975 H. Friedman introduced two statements of hyperarithmetic analysis, (sequential limit system) and (arithmetic Bolzano-Weierstrass), which are motivated by standard and well-known theorems from analysis such as the Bolzano-Weierstrass theorem for and sets of reals. In this article we characterize the reverse mathematical strength of by comparing it to most known theories of hyperarithmetic analysis.

In particular we show that, over , is equivalent to , and that is implied by , and implies . We then use Steel's method of forcing with tagged trees to show that is incomparable with (i.e. Jullien's Theorem) and . This makes the first theory of hyperarithmetic analysis that is known to be incomparable with other (known) theories of hyperarithmetic analysis. We also examine the reverse mathematical strength of the Bolzano-Weierstrass theorem in the context of open, closed, , , and other types of sets.

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Additional Information

**Chris J. Conidis**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Address at time of publication:
Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

DOI:
https://doi.org/10.1090/S0002-9947-2012-05416-X

Received by editor(s):
September 18, 2009

Received by editor(s) in revised form:
July 7, 2010

Published electronically:
April 18, 2012

Additional Notes:
The author was partially supported by NSERC grant PGS D2-344244-2007. Furthermore, he would like to acknowledge the helpful input he received from his thesis advisors: R.I. Soare, D.R. Hirschfeldt, and A. Montalbán.

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.