Aspects of the Real Numbers: Putnam, Wittgenstein, and Nonextensionalism

The Monist ◽  
2020 ◽  
Vol 103 (4) ◽  
pp. 427-441
Author(s):  
Juliet Floyd
Keyword(s):  
Set Theory ◽  
Philosophy Of Language ◽  
Real Numbers ◽  
The Real ◽  
Philosophy Of Computation ◽  
The World ◽  
Structuralist View

Abstract I defend Putnam’s modal structuralist view of mathematics but reject his claims that Wittgenstein’s remarks on Dedekind, Cantor, and set theory are verificationist. Putnam’s “realistic realism” (1990–2016) showcases the plasticity of our “fitting” words to the world. The applications of this—in philosophy of language, mind, logic, and philosophy of computation—are robust. I defend Wittgenstein’s nonextensionalist understanding of the real numbers, showing how it fits Putnam’s view. Nonextensionalism and extensionalism about the real numbers are mathematically, philosophically, and logically robust, but the two perspectives are often confused with one another. I separate them, using Turing’s work as an example.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

Towards a consistent set-theory

1951 ◽  
Vol 16 (2) ◽  
pp. 130-136 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Set Theory ◽  
Mathematical Practice ◽  
Axiom Of Choice ◽  
Small Fragment ◽  
Real Numbers ◽  
The Real ◽  
Classical Construction

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

Sets and Numbers

2021 ◽  
pp. 3-27
Author(s):  
James Davidson
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Euclidean Distance ◽  
Final Section ◽  
Real Analysis ◽  
Real Numbers ◽  
The Real ◽  
Basic Ideas ◽  
Distance Sets

This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance, sets of the real line, coverings, and compactness.

Cantor and Continuity

2020 ◽  
pp. 219-254
Author(s):  
Akihiro Kanamori
Keyword(s):  
19Th Century ◽  
Set Theory ◽  
Mathematical Analysis ◽  
Measure Theory ◽  
Georg Cantor ◽  
Real Numbers ◽  
The Real ◽  
Connected Sets

Georg Cantor (1845-1919) made seminal contributions to the mathematical conceptualization of continuity and continua that would become basic for the development of topology and measure theory in mathematics. His articulations in this direction were part and parcel of his development of set theory out of mathematical analysis and, on a larger canvas, very much part of the rigorization of mathematics in the latter 19th Century. We consider Cantor’s work on the formulation of the real numbers; uncountability and dimension; and continua as formulated in terms of perfect and connected sets as seen in this light. The thematic emphasis will be on the drive of mathematical necessity for the mathematization of metaphysicially based concepts.

Exploring Citizenship: Unit 4 My Heritage

EDIS ◽  
2019 ◽  
Vol 2019 (5) ◽  
pp. 14
Author(s):  
John Rutledge ◽  
Joy C. Jordan ◽  
Dale W. Pracht
Keyword(s):  
Youth Development ◽  
Real Life ◽  
Development Program ◽  
The Real ◽  
Volunteer Leaders ◽  
Youth Development Program ◽  
Major Revision ◽  
The World ◽  
Individual Needs

 The 4-H Citizenship Project offers the opportunity to help 4-H members relate all of their 4-H projects and experiences to the world around them. The 4-H Citizenship manuals will serve as a guide for 4-H Citizenship experiences. To be truly meaningful to the real-life needs and interests of your group, the contribution of volunteer leaders is essential. Each person, neighborhood, and community has individual needs that you can help your group identify. This 14-page major revision of Unit IV covers the heritage project. Written by John Rutledge, Joy C. Jordan, and Dale Pracht and published by the UF/IFAS Extension 4-H Youth Development program. https://edis.ifas.ufl.edu/4h019

Risk and Return of Retail Sukuk and Retail Bond in Indonesia Period 2008-2017 (Comparative Study)

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Rahma Yudi Astuti ◽  
Asad Arsya Brilliant Fani
Keyword(s):  
Standard Deviation ◽  
Comparative Study ◽  
Stock Exchange ◽  
Secondary Data ◽  
Bond Market ◽  
Bond Pricing ◽  
Risk And Return ◽  
Underlying Asset ◽  
The Real ◽  
The World

Sukuk and Bonds has differences and similarities. Fundamental differences between sukuk and bonds are first, underlying asset in every sukuk issuance, concept of profit loss sharing and the use of Islamic contracts. Whereas conducted research in practice of differences between sukuk and bonds are still an on-going discussion. This study aims to add the evidence in the discussion regarding whether there is differences between sukuk and bonds in the world of practice, provide investment preferences as well as educating investors in choosing sukuk or bonds as a sustainable and smooth instrument. The method used is Mann Whitney U-Test to test whether there is a different between yield to maturity (return) and standard deviation (risk) of both instruments. Using secondary data of Retail Sukuk (SR) and Retail Bonds (ORI) period 2008-2017 obtained from Indonesia Stock Exchange, Indonesia Bond Market Directory and Indonesia Bond Pricing Agency. The result shows that there is no significance difference of retail sukuk return and risk with retail bonds in Indonesia. Besides retail bonds are show higher return than retail sukuk because of higher coupon and longest mature date. While, retail sukuk is more stable rather than bonds as it backed up by the real underlying asset. Keywords: Retail Sukuk (SR), Retail Bonds (ORI), Yield to Maturity

The Supply-Demand Research of Cultural and Tourism Industries Policies in China with the Impact of COVID-19

2020 ◽  
Vol 11 (1) ◽  
pp. 24-24
Author(s):  
Dr. Jianfei Yang
Keyword(s):  
Tourism Industry ◽  
Chinese Government ◽  
Cultural Industries ◽  
Real Situation ◽  
The Real ◽  
The World ◽  
The Impact ◽  
Near Future ◽  
Post Period

COVID-19 has made a bad influence on economic and society including cultural and tourism industry in China，2020.The industry has received a huge loss in the first quarter of the year and the situation is getting worse in the near future. It is believed that there will be a long impact for the country even the world. In order to recover the industry, Chinese government has published series of policies to support the enterprises and clusters to reduce the bad influence of COVID-19. This paper mainly uses filed survey and documentary research to map the real situation of the industry. It tries to find the policy demand of the industries and then analyze the policies published by government to conquer COVID-19. Meanwhile it will focus on whether the supply meet the demand and give suggestions on how to promote the policy efficiency in the post period of COVID-19 in China. Keywords: Evaluation; Cultural Industries; Policy; Park; Pandemic

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