scholarly journals ENUMERABILITY OF THE SET OF ALGEBRAIC NUMBERS

2021 ◽  
Vol 12 (3) ◽  
pp. 71-77
Author(s):  
Fernanda Loureiro Honorio ◽  
Fernando Pereira de Souza
Keyword(s):  
Set Theory ◽  
Algebraic Numbers ◽  
Basic Concepts ◽  
Infinite Sets

Through the studies of the Set Theory, the cardinality of the infinite sets is classified according to their enumerability, thus making the importance of the theme remarkable. Therefore, in this work, the enumerability of the Set of Algebraic Numbers will be approached, through studies that were carried out as part of the scientific initiation project linked as activities of the group PET Connections of Knowledge Mathematics at UFMS, Campus of Três Lagoas. The concepts defined for the study of the AlgebraicNumbers Set will be conceptualized, exposing their definitions and properties, with examples and accounts of their theorems. This work, besides aiming to deepen the studies on the theme, also aims, in the presentation of the same, a way to make its understanding more accessible. In this way, it is found in the results of this research the exposure of basic concepts about the enumerability of sets, which facilitate the understanding of the most complex counts of the propositions that follow.

Dedekind, Julius Wilhelm Richard (1831–1916)

2018 ◽  
Author(s):  
Howard Stein
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Mathematical Knowledge ◽  
Mathematical Structure ◽  
Georg Cantor ◽  
General Notion ◽  
Algebraic Numbers ◽  
Integral Calculus ◽  
Foundational Work ◽  
First Time

Dedekind is known chiefly, among philosophers, for contributions to the foundations of the arithmetic of the real and the natural numbers. These made available for the first time a systematic and explicit way, starting from very general notions (which Dedekind himself regarded as belonging to logic), to ground the differential and integral calculus without appeal to geometric ‘intuition’. This work also forms a pioneering contribution to set theory (further advanced in Dedekind’s correspondence with Georg Cantor) and to the general notion of a ‘mathematical structure’. Dedekind’s foundational work had a close connection with his advancement of substantive mathematical knowledge, particularly in the theories of algebraic numbers and algebraic functions. His achievements in these fields make him one of the greatest mathematicians of the nineteenth century.

INNER MODEL THEORETIC GEOLOGY

2016 ◽  
Vol 81 (3) ◽  
pp. 972-996 ◽  
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Set Theory ◽  
Main Theme ◽  
Solid Core ◽  
Strong Cardinal ◽  
The Third ◽  
Inner Models ◽  
Inner Model ◽  
Basic Concepts ◽  
Woodin Cardinal ◽  
The Universe

AbstractOne of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P-constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ω (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model.

Application of Rough Set Theory to Data Mining of Condenser Diagnosis in Power Plants

2003 ◽  
Author(s):  
Zhongguang Fu ◽  
Tao Jin ◽  
Kun Yang
Keyword(s):  
Data Mining ◽  
Set Theory ◽  
Rough Set ◽  
Power Plants ◽  
Rough Set Theory ◽  
A Priori ◽  
Reduction Algorithm ◽  
Useful Knowledge ◽  
Fault Features ◽  
Basic Concepts

Rough set theory is a powerful tool in deal with vagueness and uncertainty. It is particularly suitable to discover hidden and potentially useful knowledge in data and can be used to reduce features and extract rules. This paper introduces the basic concepts and fundamental elements of the rough set theory. A reduction algorithm that integrates a priori with significance is proposed to illustrate how the rough set theory could be used to extract fault features of the condenser in a power plant. Two testing examples are then presented to demonstrate the effectiveness of the theory in fault diagnosis.

scholarly journals Topology in the Alternative Set Theory and Rough Sets via Fuzzy Type Theory

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 432 ◽  
Author(s):  
Vilém Novák
Keyword(s):  
Fuzzy Logic ◽  
Set Theory ◽  
Rough Set ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Rough Sets ◽  
Type Theory ◽  
Rough Set Theory ◽  
Basic Concepts ◽  
Alternative Set

In this paper, we will visit Rough Set Theory and the Alternative Set Theory (AST) and elaborate a few selected concepts of them using the means of higher-order fuzzy logic (this is usually called Fuzzy Type Theory). We will show that the basic notions of rough set theory have already been included in AST. Using fuzzy type theory, we generalize basic concepts of rough set theory and the topological concepts of AST to become the concepts of the fuzzy set theory. We will give mostly syntactic proofs of the main properties and relations among all the considered concepts, thus showing that they are universally valid.

scholarly journals Fuzzy Insurance

1990 ◽  
Vol 20 (1) ◽  
pp. 33-55 ◽  
Author(s):  
Jean Lemaire
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Life Insurance ◽  
Fuzzy Set Theory ◽  
Fuzzy Numbers ◽  
Fuzzy Decision Making ◽  
Fuzzy Decision ◽  
Basic Concepts ◽  
Definition Of

AbstractFuzzy set theory is a recently developed field of mathematics, that introduces sets of objects whose boundaries are not sharply defined. Whereas in ordinary Boolean algebra an element is either contained or not contained in a given set, in fuzzy set theory the transition between membership and non-membership is gradual. The theory aims at modelizing situations described in vague or imprecise terms, or situations that are too complex or ill-defined to be analysed by conventional methods. This paper aims at presenting the basic concepts of the theory in an insurance framework. First the basic definitions of fuzzy logic are presented, and applied to provide a flexible definition of a “preferred policyholder” in life insurance. Next, fuzzy decision-making procedures are illustrated by a reinsurance application, and the theory of fuzzy numbers is extended to define fuzzy insurance premiums.

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

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.

scholarly journals Small sets with large power sets

1973 ◽  
Vol 8 (3) ◽  
pp. 413-421 ◽  
Author(s):  
G.P. Monro
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Large Power ◽  
Present Evidence ◽  
Power Set ◽  
Infinite Sets ◽  
The Axiom Of Choice

One problem in set theory without the axiom of choice is to find a reasonable way of estimating the size of a non-well-orderable set; in this paper we present evidence which suggests that this may be very hard. Given an arbitrary fixed aleph κ we construct a model of set theory which contains a set X such that if Y ⊆ X then either Y or X - Y is finite, but such that κ can be mapped into S(S(S(X))). So in one sense X is large and in another X is one of the smallest possible infinite sets. (Here S(X) is the power set of X.)

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