Predicative aspects of order theory in univalent foundations

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Abstract

We investigate predicative aspects of order theory in constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky’s propositional resizing axioms or excluded middle. Our work complements existing work on predicative mathematics by exploring what cannot be done predicatively in univalent foundations. Our first main result is that nontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivial poset is small, then weak propositional resizing holds. It is possible to derive full propositional resizing if we strengthen nontriviality to positivity. The distinction between nontriviality and positivity is analogous to the distinction between nonemptiness and inhabitedness. We prove our results for a general class of posets, which includes directed complete posets, bounded complete posets and sup-lattices, using a technical notion of a δ_V-complete poset. We also show that nontrivial locally small δ_V-complete posets necessarily lack decidable equality. Specifically, we derive weak excluded middle from assuming a nontrivial locally small δ_V-complete poset with decidable equality. Moreover, if we assume positivity instead of nontriviality, then we can derive full excluded middle. Secondly, we show that each of Zorn’s lemma, Tarski’s greatest fixed point theorem and Pataraia’s lemma implies propositional resizing. Hence, these principles are inherently impredicative and a predicative development of order theory must therefore do without them. Finally, we clarify, in our predicative setting, the relation between the traditional definition of sup-lattice that requires suprema for all subsets and our definition that asks for suprema of all small families.
Original languageEnglish
Title of host publication6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)
EditorsNaoki Kobayashi
PublisherSchloss Dagstuhl
Number of pages18
ISBN (Electronic)9783959771917
DOIs
Publication statusPublished - 6 Jul 2021
Event6th International Conference on Formal Structures for Computation and Deduction
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Duration: 17 Jul 202124 Jul 2021
https://fscd2021.dc.uba.ar/

Publication series

NameLIPIcs: Leibniz International Proceedings in Informatics
PublisherSchloss Dagstuhl
Volume195
ISSN (Electronic)1868-8969

Conference

Conference6th International Conference on Formal Structures for Computation and Deduction
Abbreviated titleFSCD 2021
Period17/07/2124/07/21
Internet address

Bibliographical note

Publisher Copyright:
© 2021 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

  • Constructivity
  • Order theory
  • Predicativity
  • Univalent foundations

ASJC Scopus subject areas

  • Software

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