The Mathematical Development of Set Theory from Cantor to Cohen

1996 ◽  
Vol 2 (1) ◽  
pp. 1-71 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Opposite Direction ◽  
Consistency Strength ◽  
Mathematical Development ◽  
Mathematical Structures ◽  
The Creation ◽  
Historical Heritage ◽  
The Continuum

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem.

Gödel's Conceptual Realism

2005 ◽  
Vol 11 (2) ◽  
pp. 207-224 ◽  
Author(s):  
Donald A. Martin
Keyword(s):  
Set Theory ◽  
Bertrand Russell ◽  
Sense Perception ◽  
Sense Experience ◽  
Mathematical Intuition ◽  
Conceptual Realism ◽  
Kurt Gödel ◽  
Real Objects ◽  
The Continuum

Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.The first statement is a platonist declaration of a fairly standard sort concerning set theory. What is unusual in it is the inclusion of concepts among the objects of mathematics. This I will explain below. The second statement expresses what looks like a rather wild thesis.

Development of Cultural and Historical Tourism in the Regions of Russia (On the Example of the Siberian Federal District) and France (State Regulation): Features of Financial Support

2021 ◽  
pp. 124-135
Author(s):  
Dominique Crozat ◽  
◽  
Tatiana V. Zakharova ◽  
Yulia V. Podoprigora ◽  
◽  
...  
Keyword(s):  
Federal District ◽  
State Regulation ◽  
University Leadership ◽  
Museum Exhibits ◽  
Economic Role ◽  
Theatrical Performances ◽  
History Of ◽  
Local Residents ◽  
The Creation ◽  
Historical Heritage

The aim of the article is to show the history of preservation of cultural and historical heritage and the conditions for sustainable development of modern university cities on the example of university and other museums in Russia (mainly Siberia) and France. The main objective is to demonstrate the creation of harmonious campuses, in which economic, environmental and social principles are balanced, serving, among other things, to attract tourists, which at the same time contributes to the creation of new jobs. Using examples, it is considered how, with the help of coordinated actions of local authorities and university leadership, it is possible to resolve the eternal conflict between tourists and local residents. The development of regional museums and the formation of excursions for museum-educational-historical tourism of the Siberian region are analyzed. The analysis of digitization of museum exhibits and opening of access to collections to visitors from all over the world is shown. All this makes it possible to give museum lessons, to conduct toponymic excursions (scientists’ names of in street names), field games using local history and traditions, and to introduce elements of theatrical performances. The article shows how virtual tours are organized taking into account international examples. The economic role of museums, festivals, exhibitions in the life of the region is highlighted.

scholarly journals THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

2019 ◽  
Vol 85 (1) ◽  
pp. 338-366 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
SANDRA MÜLLER
Keyword(s):  
Set Theory ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

scholarly journals SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

2018 ◽  
Vol 83 (04) ◽  
pp. 1512-1538 ◽  
Author(s):  
CHRIS LAMBIE-HANSON ◽  
PHILIPP LÜCKE
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Consistency Strength ◽  
Chain Conditions ◽  
Regular Cardinals ◽  
Aronszajn Tree ◽  
Square Principles ◽  
Image Position ◽  
Aronszajn Trees ◽  
Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

FUZZY SETS AND FUZZY RELATIONAL STRUCTURES AS CHU SPACES

2000 ◽  
Vol 08 (04) ◽  
pp. 471-479 ◽  
Author(s):  
BASIL K. PAPADOPOULOS ◽  
APOSTOLOS SYROPOULOS
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Linear Logic ◽  
Relational Structure ◽  
Mathematical Structures ◽  
Relational Structures ◽  
Chu Spaces

Chu spaces, which derive from the Chu construct of *-autonomous categories, can be used to represent most mathematical structures. Moreover, the logic of Chu spaces is linear logic. Most efforts to incorporate fuzzy set theory into the realm of linear logic are based on the assumption that fuzzy and linear negation are identical operations. We propose an incorporation based on the opposite assumption and we provide an interpretation of some linear connectives. Furthermore, we show that it is possible to represent any fuzzy relational structure as a Chu space by means of the functor G.

Historically Speaking—: The Early Beginnings of Set Theory

1970 ◽  
Vol 63 (8) ◽  
pp. 690-692
Author(s):  
Phillip E. Johnson
Keyword(s):  
Present Article ◽  
Set Theory ◽  
Euclidean Geometry ◽  
Non Euclidean Geometry ◽  
The Creation

Georg Canttor created and largely developed the theory of sets in approximately the year. 1874-1897. In contrast to such developments as the calculus and non-Euclidean geometry, The creation of set theory was, according to all indications, Cantor's alone. AIso, set theory was not preceded by a long evoIutionary period such as is usually the case with big mathematical breakthroughs. The present article will concern itself primarily with the very earliest set-theoretic works of Cantor, namely, his first two papers in this area.1

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