Abstract
We study a transfinite construction we call tower construction in classical type theory. The construction is inductive and applies to partially ordered types. It yields the set of all points reachable from a starting point with an increasing successor function and a family of admissible suprema. Based on the construction, we obtain type-theoretic versions of the theorems of Zermelo (well-orderings), Hausdorff (maximal chains), and Bourbaki and Witt (fixed points). The development is formalized in Coq assuming excluded middle.
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Notes
- 1.
Our initial motivation for studying the general tower construction was the direct construction of the cumulative hierarchy in axiomatic set theory. Only after finishing the proofs for the general tower construction in type theory, we discovered Bourbaki’s marvelous presentation [2] of the tower construction in set theory.
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Acknowledgement
It was Chad E. Brown who got us interested in the topic of this paper when in February 2014 he came up with a surprisingly small Coq formalization of Zermelo’s second proof of the well-ordering theorem using an inductive definition for the least \(\varTheta \)-chain. In May 2015, Frédéric Blanqui told us about the papers of Felscher and Hessenberg in Tallinn.
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Smolka, G., Schäfer, S., Doczkal, C. (2015). Transfinite Constructions in Classical Type Theory. In: Urban, C., Zhang, X. (eds) Interactive Theorem Proving. ITP 2015. Lecture Notes in Computer Science(), vol 9236. Springer, Cham. https://doi.org/10.1007/978-3-319-22102-1_26
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DOI: https://doi.org/10.1007/978-3-319-22102-1_26
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