“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and (...) relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

In the sole extended break from his life and varing in this way we can associate a sysied career in England, Alan Turing spent the tem of logic with any constructive ordinal. It may be asked whether such a years 1936–1938 doing graduate work at..

CHAPTER ONE INTRODUCTION S 1.1: THE AIMS OF THIS BOOK ... God has not been so sparing to men to make them barely two- legged creatures, and left it to ...

The crux of the dispute between formalism and intuitionism, it is held, is not whether certain entities exist or not, but how the term function shall be used in mathematics. The identification of effective definition with general recursion fails because an undefined function lies concealed beneath the requirement of a finite number of substitutions, and a fresh characterization of effective definition is sought in terms of a hierarchy of ordinal recursions.A correspondence exists between primitive recursive properties and direct proofs, of (...) irrationality and transcendence for instance, and between general recursive properties and indirect proofs.Mathematics is a concept creating activity and the distinction between a formal mathematics devoid of meaning, at one level, and a meaningful metamathematics at the next is considered to be untenable.RésuméLa controverse entre les formalistes et les intuitionistes ne repose pas, considére‐t‐on, sur l'existence ou la non‐existence de certaines entités, mail sur l'usage du terme fonction en mathématiques. L'identification de la définition effective avec la récursion générate n'est pas valable parce qu'une fonction indéfinié est dissimulée sous la nécessité d'un nombre fini de substitutions, et une nouvelle caractérisation de définition effective est recherchée sous forme de hiérarchie de récursions ordinates.II existe, d'une part, une correspondance entre les propriétés primitives récursives et les preuves directes, d'irrationalité et de transcendance par exemple, et, d'autre part, entre les propriétés récursives générates et les preuves par l'absurde.Les mathématiques sont une activité créatrice de concepts, et la distinction entre les mathématiques formelles dépourvues de sens et, sur le plan supérieur, une méta‐mathématique chargée de sens, ne semble pas justifiée. (shrink)

This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

Alberto Cortes, in [1], attempts to show that Leibniz's Principle of The Identity of Indiscernibles is a principle restricted to individuals, and that photons appear to violate L. L is stated by Leibniz as “no two substances are completely similar, or differ solo numero.” In second-order quantification theory with identity L becomes.