Report on some investigations concerning the consistency of the axiom of reducibility

1951 ◽  
Vol 16 (1) ◽  
pp. 35-42 ◽  
Author(s):  
John Myhill
Keyword(s):  
Point Of View ◽  
Quantification Theory ◽  
Classical Mathematics

The purpose of this paper is to construct a non-finitary system of logic S, based on the theory of types, in which classical quantification theory holds without restriction and the axiom of reducibility holds with such slight restriction as is perhaps unlikely to interfere with the construction of classical mathematics. The system will be shown to be consistent.For purposes of orientation, a rough description of the ontology of the system will first be given. The ‘individuals’ are expressions built up in the following way: ‘0’ is an individual, and if ϕ and ψ are individuals, so is ˹*ϕψ˺. The usual apparatus of truth functional connectives and quantifiers is employed, and one primitive four place predicate ‘S’. ˹Sϕψχω˺ says that ϕ is the result of writing ψ for all occurrences of χ in ω, where ϕ, ψ, χ and ω may be formulae, individuals, or members of a wider category called ‘expressions’ which includes individuals, formulae, and other things as well. (At first sight this looks like a confusion of use and mention, but the formation rules and rules of inference are so phrased that no ambi guity arises.)The individuals are divided into types concentrically, so that type 0 has them all as members and type n includes type m for m > n. There is no highest type, but we shall find it convenient from the point of view of exposition to construct an infinity of ‘auxiliary’ systems Q(k), restricted to types ≦ k, before presenting the system S.

Category theory, applications to the foundations of mathematics

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Classical Logic ◽  
Elementary Theory ◽  
Mathematical Structure ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Logical Foundations ◽  
Classical Mathematics ◽  
The 1960S

Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue. Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics. Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics. Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical. This article assumes knowledge of category theory on the level of Category theory, introduction to §1.

Kronecker, Leopold (1823–91)

2018 ◽  
Author(s):  
Ulrich Majer
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Point Of View ◽  
Academic Institution ◽  
Late Nineteenth Century ◽  
Foundations Of Mathematics ◽  
Epistemological Perspective ◽  
Classical Mathematics ◽  
Late Nineteenth

Leopold Kronecker was one of the most influential German mathematicians of the late nineteenth century. He exercised a strong sociopolitical influence on the development of mathematics as an academic institution. From a philosophical point of view, his main significance lies in his anticipation of a new and rigorous epistemological perspective with regard to the foundations of mathematics: Kronecker became the father of intuitionism or constructivism, which stands in strict opposition to the methods of classical mathematics and their canonization by set theory.

scholarly journals Digital Didactics under Distance Learning Constraints

2021 ◽  
pp. 154-164
Author(s):  
N. P. Puchkov
Keyword(s):  
Numerical Analysis ◽  
Point Of View ◽  
Academic Disciplines ◽  
Mathematical Tasks ◽  
Contextual Approach ◽  
Classical Mathematics ◽  
Analytical Research ◽  
Methodological Approaches ◽  
Learning Constraints ◽  
Effective Use

The article considers methodological approaches to the process of eliminating the problems of digitalization of education using the example of the academic disciplines of mathematics and computer science. It is shown that the use of specially designed complex mathematical tasks provides a harmonious combination of analytical research inherent in classical mathematics and constantly progressing methods of numerical analysis and computer modeling. The substantive filling of educational tasks with elements of production situations from the future profession of students or from the process of their training on the principles of a contextual approach has been substantiated. The essence of the ongoing process of digitalization of education and its effective use in the context of the limitations of contact work with students is considered from a constructive point of view.

A STRONG MULTI-TYPED INTUITIONISTIC THEORY OF FUNCTIONALS

2015 ◽  
Vol 80 (3) ◽  
pp. 1035-1065 ◽  
Author(s):  
FARIDA KACHAPOVA
Keyword(s):  
Set Theory ◽  
Classical Theory ◽  
Second Order ◽  
Point Of View ◽  
Second Order Arithmetic ◽  
Intuitionistic Theory ◽  
Classical Mathematics ◽  
Order Arithmetic ◽  
Comprehension Axiom

AbstractIn this paper we describe an intuitionistic theory SLP. It is a relatively strong theory containing intuitionistic principles for functionals of many types, in particular, the theory of the “creating subject”, axioms for lawless functionals and some versions of choice axioms. We construct a Beth model for the language of intuitionistic functionals of high types and use it to prove the consistency of SLP.We also prove that the intuitionistic theory SLP is equiconsistent with a classical theory TI. TI is a typed set theory, where the comprehension axiom for sets of type n is restricted to formulas with no parameters of types > n. We show that each fragment of SLP with types ≤ s is equiconsistent with the corresponding fragment of TI and that it is stronger than the previous fragment of SLP. Thus, both SLP and TI are much stronger than the second order arithmetic. By constructing the intuitionistic theory SLP and interpreting in it the classical set theoryTI, we contribute to the program of justifying classical mathematics from the intuitionistic point of view.

scholarly journals Sistemas de cálculo como formas de Logicismo

1993 ◽  
Vol 25 (73) ◽  
pp. 15-35
Author(s):  
Ángel Nepomuceno Fernández
Keyword(s):  
Computer Science ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Common Language ◽  
Philosophy Of Logic ◽  
Classical Mathematics ◽  
And Mathematics ◽  
The One ◽  
Modal Systems ◽  
Modified Functional

The logicism may be regarded like a fossil stone that has not utility nowadays. In this sense, logicism took care of the research about the foundations of mathematics but apparently its task arrived at its end many years ago because of sorne results that were eetablished during the century. However it is not wholly right. Understanding logicism as an attempt to reduce classical mathematics to logic means we can distinguish: 1) the idea according to which mathematic is logic in sorne way, and 2) a metaphysical program of research to: a) define mathematical notions as logical notions, and b) show that the mathematical theorems are logical theorems. The failure (if so) concerned to 2), since 1) was assumed by many logicians. Recovering logicism is not easy and there may be several ways. One of them is the one followed by N.B. Cocchiarella whose systems (there are more than one) represent a form of logicism (Frege's or Russell's form). From those systems -though a bit changed from my own point of view- we can define a modal calcule that may have application in computer science, what would not be a stale work. From a common language we take in account two systems in order to show that Cocchiarella's modified system is as powerful deductively as that of Church modified functional second order calcule. We can obtain new systems that represent form of logicism and are more powerful than that of Church enlarging Cochiarella's modified system. These new systems, that becomes modal systems provided that one adds appropiate modal tools (then they may be used in computer science), may be useful to study logicism itself (as historical philosophy of logic and mathematics).

The Geosciences Applied to Lunar Exploration

1962 ◽  
Vol 14 ◽  
pp. 169-257 ◽  
Author(s):  
J. Green
Keyword(s):  
Lunar Surface ◽  
Probable Mechanism ◽  
Point Of View ◽  
Lunar Exploration ◽  
Geological Model ◽  
Apply Geophysics ◽  
Surface Features ◽  
The Impact

The term geo-sciences has been used here to include the disciplines geology, geophysics and geochemistry. However, in order to apply geophysics and geochemistry effectively one must begin with a geological model. Therefore, the science of geology should be used as the basis for lunar exploration. From an astronomical point of view, a lunar terrain heavily impacted with meteors appears the more reasonable; although from a geological standpoint, volcanism seems the more probable mechanism. A surface liberally marked with volcanic features has been advocated by such geologists as Bülow, Dana, Suess, von Wolff, Shaler, Spurr, and Kuno. In this paper, both the impact and volcanic hypotheses are considered in the application of the geo-sciences to manned lunar exploration. However, more emphasis is placed on the volcanic, or more correctly the defluidization, hypothesis to account for lunar surface features.

How a sheperd guides a sheep herd?

1984 ◽  
Vol 75 ◽  
pp. 331-337
Author(s):  
Richard Greenberg
Keyword(s):  
Point Of View ◽  
Single Ring ◽  
Damping Effects ◽  
Ring Particle

ABSTRACTThe mechanism by which a shepherd satellite exerts a confining torque on a ring is considered from the point of view of a single ring particle. It is still not clear how one might most meaningfully include damping effects and other collisional processes into this type of approach to the problem.

The growth of mica polytypes as studied by conventional and high-resolution TEM

1983 ◽  
Vol 41 ◽  
pp. 174-177
Author(s):  
A. Baronnet ◽  
M. Amouric
Keyword(s):  
Water Pressure ◽  
Point Of View ◽  
Temperature Fields ◽  
Natural Occurrence ◽  
Growth Mechanisms ◽  
Long Period ◽  
High Resolution Tem ◽  
Layer Stacking ◽  
Experimental Works ◽  
Potential Use

The origin of mica polytypes has long been a challenging problem for crystal- lographers, mineralogists and petrologists. From the petrological point of view, interest in this field arose from the potential use of layer stacking data to furnish further informations about equilibrium and/or kinetic conditions prevailing during the crystallization of the widespread mica-bearing rocks. From the compilation of previous experimental works dealing with the occurrence domains of the various mica "polymorphs" (1Mr, 1M, 2M1, 2M2 and 3T) within water-pressure vs temperature fields, it became clear that most of these modifications should be considered as metastable for a fixed mica species. Furthermore, the natural occurrence of long-period (or complex) polytypes could not be accounted for by phase considerations. This highlighted the need of a more detailed kinetic approach of the problem and, in particular, of the role growth mechanisms of basal faces could play in this crystallographic phenomenon.

Radiation Damage in Ceramics

1982 ◽  
Vol 40 ◽  
pp. 600-603
Author(s):  
T. E. Mitchell ◽  
M. R. Pascucci ◽  
R. A. Youngman
Keyword(s):  
Radiation Damage ◽  
Outer Space ◽  
Point Of View ◽  
Nuclear Waste Storage ◽  
Waste Storage ◽  
Storage Media ◽  
Transmission Electron ◽  
Fundamental Point ◽  
Small Defect ◽  
Present Understanding

1. Introduction. Studies of radiation damage in ceramics are of interest not only from a fundamental point of view but also because it is important to understand the behavior of ceramics in various practical radiation enyironments- fission and fusion reactors, nuclear waste storage media, ion-implantation devices, outer space, etc. A great deal of work has been done on the spectroscopy of point defects and small defect clusters in ceramics, but relatively little has been performed on defect agglomeration using transmission electron microscopy (TEM) in the same kind of detail that has been so successful in metals. This article will assess our present understanding of radiation damage in ceramics with illustrations using results obtained from the authors' work.

Structural aspects of osmoregulation in porphyra

1978 ◽  
Vol 36 (2) ◽  
pp. 412-413
Author(s):  
C. Wiencke ◽  
A. Lauchli
Keyword(s):  
Culture Medium ◽  
Point Of View ◽  
Chemical Fixation ◽  
Cellular Organization ◽  
Osmotic Adaptation ◽  
Osmotic Balance ◽  
Activity Of Enzymes ◽  
Osmotic Agents ◽  
Vacuolar System ◽  
Structural Aspects

Osmoregulatory mechanisms in algae were investigated mainly from a physiological point of view (KAUSS 1977, HELLEBUST 1976). In Porphyra two osmotic agents, i. e. floridoside/isofloridoside (KAUSS 1968) and certain ions, such as K+ and Na+(EPPLEY et al. 1960) are considered for osmotic balance. Accumulations of ions (particularly Na+) in the cytoplasm during osmotic adaptation is improbable, because the activity of enzymes is generally inhibited by high ionic concentrations (FLOWERS et al. 1977).The cellular organization of Porphyra was studied with special emphasis on the development of the vacuolar system under different hyperosmotic conditions. Porphyra was cultivated at various strengths of the culture medium ASP 12 (PROVASOLI 1961) ranging from normal to 6 times concentrated (6x) culture medium. Por electron microscopy freeze fracturing was used (specimens fixed in 2% glutaraldehyde and incubated in 30% glycerol, preparation in a BALZERS BA 360 M apparatus), because chemical fixation gave poor results.

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