“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..

Early versions of satellite and radiosonde datasets suggested that the tropical surface had warmed more than the troposphere, while climate models consistently showed tropospheric amplification of surface warming in response to human-caused increases in greenhouse gases. We revisit such comparisons here using new observational estimates of surface and tropospheric temperature changes. We find that there is no longer a serious discrepancy between modeled and observed trends in the tropics. Our results contradict a recent claim that all simulated temperature trends in (...) the tropical troposphere are inconsistent with observations. This claim was based on the use of older radiosonde and satellite datasets and on two methodological errors: the neglect of observational trend uncertainties introduced by interannual climate variability and application of an inappropriate statistical “consistency test”.This emerging reconciliation of models and observations has two primary explanations. First, because of changes in the treatment of buoy and satellite information, new surface temperature datasets yield slightly reduced tropical warming relative to earlier versions. Second, recently developed satellite and radiosonde datasets now show larger warming of the tropical lower troposphere. In the case of a new satellite dataset from remote sensing systems, enhancedRSS warming is due to an improved procedure of adjusting for inter-satellite biases. When the RSS-derived tropospheric temperature trend is compared with four different observed estimates of surface temperature change, the surface warming is invariably amplified in the tropical troposphere, consistent with model results. Even if we use data from a second satellite dataset with smaller tropospheric warming than in remote sensing systems RSS, observed tropical lapse rates are not significantly different from those in all model simulations.Our results contradict a recent claim that all simulated temperature trends in the tropical troposphere and in tropical lapse rates are inconsistent with observations. This claim was based on the use of older radiosonde and satellite datasets and on two methodological errors: the neglect of observational trend uncertainties introduced by interannual climate variability and application of an inappropriate statistical “consistency test”. (shrink)

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.

This paper explores the use of advance directives in clinical dementia research. The focus is on advance consent to participation of demented patients in non-therapeutic research involving more than minimal risks and/or burdens. First, morally relevant differences between advance directives for treatment and care, and advance directives for dementia research are discussed. Then attention is paid to the philosophical issue of dementia and personal identity, and the implications for the moral authority of research advance directives. Thirdly, a number of practical (...) shortcomings of advance directives for non-therapeutic dementia research are explored and attention is paid to the role of proxies. It is concluded that upon a closer look the initial attractiveness of advance directives for dementia research is lessened, and that it is doubtful whether these instruments can compensate for the lack of subject consent in case of non-therapeutic dementia research involving more than minimal risks and/or burdens for the incompetent demented subject. (shrink)