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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.



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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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