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, the fact that these nontrivial posets are necessarily large has the important consequence that Tarski's theorem (and similar results) cannot be applied in nontrivial instances. Furthermore, we explain that generalizations of Tarski's theorem that allow for large structures are provably false by showing that the ordinal of ordinals in a univalent universe has 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.
翻译:我们调查的是建设性非虚拟基础的预言性方面。 我们通过预言性和建设性, 分别意味着我们不假定Voevodsky的推论重现轴心或排除中间线。 我们的工作补充了现有的预言性数学工作, 探索了无法在非虚拟基础上完成的预言性工作。 我们的第一个主要结果必然是非初始( 定向或约束的) 完整的直言面必然是巨大的。 这就是说, 如果这种非初始的外言面部小, 那么虚构的虚构会维持住。 如果我们加强非初始性重现轴心或排除中间的中间部分, 就可能得出完全的重现性。 我们的工作表现了本地的、 非初始性( 定向或约束的) 完全性、 预知性、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 预言、 后期、 必然造成非重要的结果。