On Small Types in Univalent Foundations

We investigate predicative aspects of 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. Moreover, we prove that locally small, nontrivial (directed or bounded) complete posets necessarily lack decidable equality. We prove our results for a general class of posets, which includes e.g. directed complete posets, bounded complete posets, sup-lattices and frames. Secondly, we discuss the unavailability of Zorn's lemma, Tarski's greatest fixed point theorem and Pataraia's lemma in our predicative setting, and prove the ordinal of ordinals in a univalent universe to have small suprema in the presence of set quotients. The latter also leads us to investigate the inter-definability and interaction of type universes of propositional truncations and set quotients, as well as a set replacement principle. Thirdly, 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.Comment: Extended version of arXiv:2102.08812. v2: Revised and expanded following referee report

On the existence of Stone-Cech compactification

In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann. Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained of the locales of which the Stone-Cech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of dependent choice. In this paper I show that this characterization continues to hold over the standard system CZF plus REA, thus removing in particular any dependency from a choice principle. This will follow by a result of independent interest, namely the proof that the class of continuous mappings from a compact regular locale X to a regular a set-presented locale Y is a set in CZF, even without REA. It is then shown that the existence of Stone-Cech compactification of a non-degenerate Boolean locale is independent of the axioms of CZF (+REA), so that the obtained characterization characterizes a proper subcollection of the collection of all locales. The same also holds for several, even impredicative, extensions of CZF+REA, as well as for CTT. This is in contrast with what happens in the context of Higher-order Heyting arithmetic HHA - and thus in any topos-theoretic universe: by constructions of Johnstone, Banaschewski and Mulvey, within HHA Stone-Cech compactification can be defined for every locale

Instance reducibility and Weihrauch degrees

We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in reverse constructive mathematics (formal Church's thesis, Brouwer's Continuity principle and Fan theorem, Excluded middle, Limited principle, Function choice, Markov's principle, etc.). We show that the instance degrees form a frame, i.e., a complete lattice in which finite infima distribute over set-indexed suprema. They turn out to be equivalent to the frame of upper sets of truth values, ordered by the reverse Smyth partial order. We study the overall structure of the lattice: the subobject classifier embeds into the lattice in two different ways, one monotone and the other antimonotone, and the $\lnot\lnot$-dense degrees coincide with those that are reducible to the degree of Excluded middle. We give an explicit formulation of instance degrees in a relative realizability topos, and call these extended Weihrauch degrees, because in Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees correspond precisely to Weihrauch degrees. The extended degrees improve the structure of Weihrauch degrees by equipping them with computable infima and suprema, an implication, the ability to control access to parameters and computation of results, and by generally widening the scope of Weihrauch reducibility

Computability of probability measures and Martin-Lof randomness over metric spaces

In this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).Comment: 29 page