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

Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics

What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem

Finiteness in cubical type theory

This thesis will explore and explain finiteness in constructive mathematics: using this setting, it will also serve as an introduction to constructive mathematics in Cubical Agda, and some related topics

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

A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint

SEPARATING FRAGMENTS OF WLEM, LPO, AND MP

Abstract. We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov's Principle, including Weak Markov's Principle, do not imply each other. §1. Introduction. At the beginning of the twentieth century, Brouwer identified a number of constructively dubious principles, which Bishop later, in his 1967 monograph Omniscience principles are commonly used to show the independence of more subject specific theorems: if a (classical) result constructively implies an omniscience principle, then it cannot be proved using constructive techniques. By separating different omniscience principles over IZF we make this task easier: if under the assumption of a classical result together with an omniscience principle we can derive a stronger omniscience principle, then we can still conclude that the classical theorem is nonconstructive. More generally, implications among these principles and theorems of mainstream mathematics have been studied for a long time. Often this is the motivation for introducing these principles (some references being provided with the principles below), and often this study is done for foundational reasons after the principles are already established (as, for instance, in In this paper we present many models, often related to each other, that separate a large number of the omniscience principles defined in terms of binary sequences and related principles. The genesis of this work was the first author's question to the second of whether Richman's LLPO n hierarchy [23] could be separated, a question about results. Since then, much interest has shifted to technique: could an argumen

Constructive Mathematics in Theory and Programming Practice

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics(BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focuses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH