Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   investigate the reverse mathematics of higher-order analogues of the theory \ATRz{} within the framework of higher order reverse mathematics developed by Kohlenbach \cite{Koh01}.  define a theory \RCAzthr, a close higher-type analogue of the classical base theory \RCAz, and show that it is essentially a conservative subtheory of Kohlenbach's base theory \RCAzo. Working over \RCAzthr,  study higher-type analogues of statements classically equivalent to \ATRz, including open and clopen determinacy, as ll as two choice principles, and prove several equivalences and separations. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel forcing; in the presentation of this result,  develop a new, more flexible framework for Steel-type forcing.
['Noah David Schweber']
I am writing a research paper on a idea Transfinite Recursion in Higher Reverse Mathematics
[{"from": "human", "value": "I am writing a research paper on a idea Transfinite Recursion in Higher Reverse Mathematics"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   investigate the reverse mathematics of higher-order analogues of the theory \\ATRz{} within the framework of higher order reverse mathematics developed by Kohlenbach \\cite{Koh01}.  define a theory \\RCAzthr, a close higher-type analogue of the classical base theory \\RCAz, and show that it is essentially a conservative subtheory of Kohlenbach's base theory \\RCAzo. Working over \\RCAzthr,  study higher-type analogues of statements classically equivalent to \\ATRz, including open and clopen determinacy, as ll as two choice principles, and prove several equivalences and separations. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel forcing; in the presentation of this result,  develop a new, more flexible framework for Steel-type forcing."}]