, Volume 192, Issue 8, pp 2437–2462 | Cite as

Transfinite recursion and computation in the iterative conception of set

  • Benjamin RinEmail author


Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles.


Set theory Infinity Recursion Replacement Iterative conception Transfinite computation Ordinal Turing machine Kreisel 



I would like to acknowledge Kyle Banick, Philip Ehrlich, Ethan Galebach, Peter Koepke, Christopher Menzel, Charles Parsons, Erich Reck, Sam Roberts, Sam Sanders, Claudio Ternullo, Sean Walsh, Kai Wehmeier, and John Wigglesworth, for their numerous helpful suggestions. I would also like to thank the referees for their many written comments. Finally, I wish to thank Benedikt Löwe, Dirk Schlimm, and the other organizers of FotFS VIII. The research for this paper was partly funded by a grant from the Analysis Trust.


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Authors and Affiliations

  1. 1.University of CaliforniaIrvineUSA

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