25 research outputs found
Gentzenから始まる証明論の50年 : 順序数解析を中心として (証明と計算の理論と応用)
おおよそ1930-80年における証明論の主な結果・アイデアを,順序数解析(ordinal analysis)を中心として述べていく.但しこの期間の問題に関わる限り,90年以降の結果も一部盛り込む.尚,記述や記法は後に整理されたかたちで述べるので原論文のままというわけではない.したがって証明論の通史や学史のようなものをこの原稿に期待しないで頂きたい.ここでは紙幅の制限により証明の詳細は省いてある.sequent calculi(とε-calucliも少々)については[A2020a]をご参照願いたい
Broad Infinity and Generation Principles
This paper introduces Broad Infinity, a new and arguably intuitive axiom
scheme. It states that "broad numbers", which are three-dimensional trees whose
growth is controlled, form a set. If the Axiom of Choice is assumed, then Broad
Infinity is equivalent to the Ord-is-Mahlo scheme: every closed unbounded class
of ordinals contains a regular ordinal.
Whereas the axiom of Infinity leads to generation principles for sets and
families and ordinals, Broad Infinity leads to more advanced versions of these
principles. The paper relates these principles under various prior assumptions:
the Axiom of Choice, the Law of Excluded Middle, and weaker assumptions.Comment: 52 page
The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences
In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines
On the correspondence of object classifiers and Tarski universes
We prove a correspondence between -small fibrations in simplicial
presheaf categories equipped with the injective or projective model structure
(and left Bousfield localizations thereof) and relative -compact maps
in their underlying quasi-categories for suitably large inaccessible cardinals
. We thus obtain a transition result between weakly universal small
fibrations in the injective Dugger-Rezk-style standard presentations of model
toposes and object classifiers in Grothendieck -toposes in the sense of
Lurie.Comment: 14 page
Partial functions and recursion in univalent type theory
We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability theory. We begin with a treatment of partial functions, using the notion of dominance, which is used in synthetic domain theory to discuss classes of partial maps. We relate this and other ideas from synthetic domain theory to other approaches to partiality in type theory. We show that the notion of dominance is difficult to apply in our setting: the set of �0 1 propositions investigated by Rosolini form a dominance precisely if a weak, but nevertheless unprovable, choice principle holds. To get around this problem, we suggest an alternative notion of partial function we call disciplined maps. In the presence of countable choice, this notion coincides with Rosolini’s. Using a general notion of partial function,we take the first steps in constructive computability theory. We do this both with computability as structure, where we have direct access to programs; and with computability as property, where we must work in a program-invariant way. We demonstrate the difference between these two approaches by showing how these approaches relate to facts about computability theory arising from topos-theoretic and typetheoretic concerns. Finally, we tie the two threads together: assuming countable choice and that all total functions N - N are computable (both of which hold in the effective topos), the Rosolini partial functions, the disciplined maps, and the computable partial functions all coincide. We observe, however, that the class of all partial functions includes non-computable
partial functions