The Pratt-Ryskamp Exchange on Set Theory, Paradox, Garciadiego and the Foundations of Computer Science

2007 ◽  
Author(s):  
John Henry Ryskamp
Keyword(s):  
Computer Science ◽  
Set Theory

An Essay on Denotational Mathematics

2019 ◽  
pp. 1629-1640
Author(s):  
Giuseppe Iurato
Keyword(s):  
Artificial Intelligence ◽  
Computer Science ◽  
Set Theory ◽  
Neuronal Networks ◽  
Theoretical Computer Science ◽  
Mathematical Framework ◽  
Theoretical Computer ◽  
Cognitive Informatics ◽  
Naive Set Theory ◽  
Software Science

Denotational mathematics is a new rigorous discipline of theoretical computer science that springs out from the attempt to provide a suitable mathematical framework in which laid out new algebraic structures formalizing certain formal patterns coming from computational and natural intelligence, software science, cognitive informatics, neuronal networks, and artificial intelligence. In this chapter, a very brief but rigorous exposition of the main formal structures of denotational mathematics is outlined within naive set theory.

scholarly journals WEIGHTED AUTOMATA AS COALGEBRAS IN CATEGORIES OF MATRICES

2013 ◽  
Vol 24 (06) ◽  
pp. 709-728 ◽  
Author(s):  
JOSÉ N. OLIVEIRA
Keyword(s):  
Computer Science ◽  
Linear Algebra ◽  
Set Theory ◽  
Matrix Algebra ◽  
Quantitative Methods ◽  
Qualitative Reasoning ◽  
Formal Logic ◽  
Quantitative Reasoning ◽  
Weighted Automata ◽  
Conventional Notation

The evolution from non-deterministic to weighted automata represents a shift from qualitative to quantitative methods in computer science. The trend calls for a language able to reconcile quantitative reasoning with formal logic and set theory, which have for so many years supported qualitative reasoning. Such a lingua franca should be typed, polymorphic, diagrammatic, calculational and easy to blend with conventional notation. This paper puts forward typed linear algebra as a candidate notation for such a unifying role. This notation, which emerges from regarding matrices as morphisms of suitable categories, is put at work in describing weighted automata as coalgebras in such categories. Some attention is paid to the interface between the index-free (categorial) language of matrix algebra and the corresponding index-wise, set-theoretic notation.

scholarly journals The Prospects for Mathematical Logic in the Twenty-First Century

2001 ◽  
Vol 7 (2) ◽  
pp. 169-196 ◽  
Author(s):  
Samuel R. Buss ◽  
Alexander S. Kechris ◽  
Anand Pillay ◽  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
First Century ◽  
Future Developments ◽  
The Future ◽  
Twenty First Century

AbstractThe four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

scholarly journals Soft rational line integral

2021 ◽  
Vol 31 (4) ◽  
pp. 578-596
Author(s):  
S. Acharjee ◽  
D.A. Molodtsov
Keyword(s):  
Computer Science ◽  
Set Theory ◽  
Soft Set ◽  
Theoretical Development ◽  
Classical Analysis ◽  
Computing Systems ◽  
Line Integral ◽  
Fundamental Properties ◽  
Necessary And Sufficient ◽  
Rational Line

Soft set theory is a new area of mathematics that deals with uncertainties. Applications of soft set theory are widely spread in various areas of science and social science viz. decision making, computer science, pattern recognition, artificial intelligence, etc. The importance of soft set-theoretical versions of mathematical analysis has been felt in several areas of computer science. This paper suggests some concepts of a soft gradient of a function and a soft integral, an analogue of a line integral in classical analysis. The fundamental properties of soft gradients are established. A necessary and sufficient condition is found so that a set can be a subset of the soft gradient of some function. The inclusion of a soft gradient in a soft integral is proved. Semi-additivity and positive uniformity of a soft integral are established. Estimates are obtained for a soft integral and the size of its segment. Semi-additivity with respect to the upper limit of integration is proved. Moreover, this paper enriches the theoretical development of a soft rational line integral and associated areas for better functionality in terms of computing systems.

On Soft Graphs and Chained Soft Graphs

2018 ◽  
Vol 7 (2) ◽  
pp. 85-102
Author(s):  
K. P. Ratheesh
Keyword(s):  
Computer Science ◽  
Set Theory ◽  
Equivalence Relation ◽  
Medical Science ◽  
Soft Set ◽  
Fuzzy Graph

Soft set theory has a rich potential for application in many scientific areas such as medical science, engineering and computer science. This theory can deal uncertainties in nature by parametrization process. In this article, the authors explore the concepts of soft relation on a soft set, soft equivalence relation on a soft set, soft graphs using soft relation, vertex chained soft graphs and edge chained soft graphs and investigate various types of operations on soft graphs such as union, join and complement. Also, it is established that every fuzzy graph is an edge chained soft graph.

The Mathematical Import of Zermelo's Well-Ordering Theorem

1997 ◽  
Vol 3 (3) ◽  
pp. 281-311 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Computer Science ◽  
Set Theory ◽  
Point Theorem ◽  
Basic Result ◽  
Subsequent Development ◽  
Power Set ◽  
Early Expression ◽  
The Subject

Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science.

scholarly journals Ramsey Theory Applications

10.37236/34 ◽  
2004 ◽  
Vol 1000 ◽  
Author(s):  
Vera Rosta
Keyword(s):  
Information Theory ◽  
Number Theory ◽  
Computer Science ◽  
Ergodic Theory ◽  
Set Theory ◽  
Ramsey Theory ◽  
Theoretical Computer Science ◽  
Theoretical Computer ◽  
Ramsey Type

There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer science. Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs. The main objective of this survey is to list applications mostly in theoretical computer science of the last two decades not contained in these.

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