Constructive set theory and Brouwerian principles

The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF

Realizability Models Separating Various Fan Theorems

Abstract. We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees. This is then applied to prove several separation results involving variants of the Fan Theorem

Lifschitz Realizability as a Topological Construction

We develop a number of variants of Lifschitz realizability for CZF by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property

Validating Brouwer's Continuity Principle for Numbers Using Named Exceptions

This paper extends the Nuprl proof assistant (a system representative of the class of extensional type theories with dependent types) withnamed exceptionsandhandlers, as well as a nominalfreshoperator. Using these new features, we prove a version of Brouwer's continuity principle for numbers. We also provide a simpler proof of a weaker version of this principle that only uses diverging terms. We prove these two principles in Nuprl's metatheory using our formalization of Nuprl in Coq and reflect these metatheoretical results in the Nuprl theory as derivation rules. We also show that these additions preserve Nuprl's key metatheoretical properties, in particular consistency and the congruence of Howe's computational equivalence relation. Using continuity and the fan theorem, we prove important results of Intuitionistic Mathematics: Brouwer's continuity theorem, bar induction on monotone bars and the negation of the law of excluded middle.</jats:p

Large sets in constructive set theory

This thesis presents an investigation into large sets and large set axioms in the context of the constructive set theory CZF. We determine the structure of large sets by classifying their von Neumann stages and use a new modified cumulative hierarchy to characterise their arrangement in the set theoretic universe. We prove that large set axioms have good metamathematical properties, including absoluteness for the common relative model constructions of CZF and a preservation of the witness existence properties CZF enjoys. Furthermore, we use realizability to establish new results about the relative consistency of a plurality of inaccessibles versus the existence of just one inaccessible. Developing a constructive theory of clubs, we present a characterisation theorem for Mahlo sets connecting classical and constructive approaches to Mahloness and determine the amount of induction contained in the assertion of a Mahlo set. We then present a characterisation theorem for 2-strong sets which proves them to be equivalent to a logically simpler concept. We also investigate several topics connected to elementary embeddings of the set theoretic universe into a transitive class model of CZF, where considering different equivalent classical formulations results in a rich and interconnected spectrum of measurability for the constructive case. We pay particular attention to the question of cofinality of elementary embeddings, achieving both very strong cofinality properties in the case of Reinhardt embeddings and constructing models of the failure of cofinality in the case of ordinary measurable embeddings, some of which require only surprisingly low conditions. We close with an investigation of constructive principles incompatible with elementary embeddings