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TRANSFINITE RECURSION IN HIGHER REVERSE MATHEMATICS

Published online by Cambridge University Press:  22 July 2015

NOAH SCHWEBER
NOAH SCHWEBER*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CALIFORNIA 94720, USAE-mail: schweber@math.berkeley.edu
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Abstract

In this paper we investigate the reverse mathematics of higher-order analogues of the theory $$ATR_0$$ within the framework of higher order reverse mathematics developed by Kohlenbach [11]. We define a theory $$RCA_0^3$$, a close higher-type analogue of the classical base theory $$RCA_0$$ which is essentially a conservative subtheory of Kohlenbach’s base theory $$RCA_{\rm{0}}^\omega$$. Working over $$RCA_0^3$$, we study higher-type analogues of statements classically equivalent to $$ATR_0$$, including open and clopen determinacy, and examine the extent to which $$ATR_0$$ remains robust at higher types. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel’s tagged tree forcing; in the presentation of this result, we develop a new, more flexible framework for Steel-type forcing.

Keywords

reverse mathematicshigher reverse mathematicsdeterminacyset theoryforcing
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 3 , September 2015 , pp. 940 - 969
Copyright
Copyright © The Association for Symbolic Logic 2015 

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