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Transactions of the American Mathematical Society

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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: In 1975 H. Friedman introduced two statements of hyperarithmetic analysis, $ \mathsf {SL_0}$ (sequential limit system) and $ \mathsf {ABW_0}$ (arithmetic Bolzano-Weierstrass), which are motivated by standard and well-known theorems from analysis such as the Bolzano-Weierstrass theorem for $ F_\sigma $ and $ G_\delta $ sets of reals. In this article we characterize the reverse mathematical strength of $ \mathsf {ABW_0}$ by comparing it to most known theories of hyperarithmetic analysis.

In particular we show that, over $ \mathsf {RCA_0+I\Sigma ^1_1}$, $ \mathsf {SL_0}$ is equivalent to $ \mathsf {\Sigma ^1_1-AC_0}$, and that $ \mathsf {ABW_0}$ is implied by $ \mathsf {\Sigma ^1_1-AC_0}$, and implies $ \mathsf {weak-\Sigma ^1_1-AC_0}$. We then use Steel's method of forcing with tagged trees to show that $ \mathsf {ABW_0}$ is incomparable with $ \mathsf {INDEC}$ (i.e. Jullien's Theorem) and $ \mathsf {\Delta ^1_1-CA_0}$. This makes $ \mathsf {ABW_0}$ 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, $ F_\sigma $, $ G_\delta $, 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.

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