Predicative Aspects of Order Theory in Univalent Foundations

TL;DR

This paper explores the limitations and implications of predicative order theory within univalent foundations, revealing that many classical principles are impredicative and cannot be assumed without enlarging the universe.

Contribution

It provides a detailed analysis of what can and cannot be achieved predicatively in univalent foundations, including results on the size of complete posets and the impredicativity of key theorems.

Findings

Nontrivial complete posets are necessarily large or imply propositional resizing.

Locally small complete posets with decidable equality lack decidable equality themselves.

Classical principles like Zorn's lemma imply propositional resizing, thus are impredicative.

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 $\delta_{\mathcal V}$-complete poset. We also show that nontrivial locally small $\delta_{\mathcal V}$-complete posets necessarily lack decidable equality. Specifically, we derive weak excluded middle from assuming a nontrivial locally small $\delta_{\mathcal V}$-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.