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.

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.

The fine structure of real mice

1998 ◽  
Vol 63 (3) ◽  
pp. 937-994 ◽  
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Fine Structure ◽  
Core Model ◽  
Structure Theory ◽  
The Real

AbstractBefore one can construct scales of minimal complexity in the Real Core Model, K(ℝ), one needs to develop the fine-structure theory of K (ℝ). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(ℝ). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(ℝ)and the strength of the theory ZF + AD + ┐DCℝ.

A complete and consistent modal set theory

1967 ◽  
Vol 32 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Continuous Functions ◽  
Subsequent Paper ◽  
Object Language ◽  
Real Numbers ◽  
Consistent System ◽  
Detailed Treatment ◽  
The Way

1.1. The aim of this paper is the construction of a demonstrably consistent system of set theory that (1) contains roughly the same amount of mathematics as the writer's system K′ [3], including a theory of continuous functions of real numbers, and (2) provides a way for expressing in the object language various propositions which, in the case of K′, could be expressed only in the metalanguage, for example, general propositions about all real numbers. It was not originally intended that the desired system should be a modal logic, but the modal character of the system appears to be a natural outgrowth of the way it is constructed. A detailed treatment of the natural, rational, and real numbers is left for a subsequent paper.

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.

Menelaah Lebih Dekat “Post Factual/Post Truth Politics, Studi Kasus Brexit” (Analsis Resensi Media)

2017 ◽  
Vol 1 (1) ◽  
Author(s):  
Eko Wahyono ◽  
Rizka Amalia ◽  
Ikma Citra Ranteallo
Keyword(s):  
United States ◽  
Mass Media ◽  
New Media ◽  
The United States ◽  
The Political ◽  
Political Struggle ◽  
The Real ◽  
The Mass Media ◽  
The Uk ◽  
The Way

This research further examines the video entitled “what is the truth about post-factual politics?” about the case in the United States related to Trump and in the UK related to Brexit. The phenomenon of Post truth/post factual also occurs in Indonesia as seen in the political struggle experienced by Ahok in the governor election (DKI Jakarta). Through Michel Foucault's approach to post truth with assertive logic, the mass media is constructed for the interested parties and ignores the real reality. The conclusion of this study indicates that new media was able to spread various discourses ranging from influencing the way of thoughts, behavior of society to the ideology adopted by a society.Keywords: Post factual, post truth, new media

The AI Delusion

2018 ◽  
Author(s):  
Gary Smith
Keyword(s):  
Data Mining ◽  
Knowledge Discovery ◽  
Industrial Revolution ◽  
The Real ◽  
Intelligent Machines ◽  
Black Boxes ◽  
Real Danger ◽  
The Way

We live in an incredible period in history. The Computer Revolution may be even more life-changing than the Industrial Revolution. We can do things with computers that could never be done before, and computers can do things for us that could never be done before. But our love of computers should not cloud our thinking about their limitations. We are told that computers are smarter than humans and that data mining can identify previously unknown truths, or make discoveries that will revolutionize our lives. Our lives may well be changed, but not necessarily for the better. Computers are very good at discovering patterns, but are useless in judging whether the unearthed patterns are sensible because computers do not think the way humans think. We fear that super-intelligent machines will decide to protect themselves by enslaving or eliminating humans. But the real danger is not that computers are smarter than us, but that we think computers are smarter than us and, so, trust computers to make important decisions for us. The AI Delusion explains why we should not be intimidated into thinking that computers are infallible, that data-mining is knowledge discovery, and that black boxes should be trusted.

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