Topological problems in alternative set theory

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

Mathematics ◽

10.3390/math8030432 ◽

2020 ◽

Vol 8 (3) ◽

pp. 432 ◽

Cited By ~ 1

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.

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THE PHILOSOPHICAL FOUNDATIONS OF ALTERNATIVE SET THEORY*

International Journal of General Systems ◽

10.1080/03081079108945019 ◽

1991 ◽

Vol 20 (1) ◽

pp. 115-126 ◽

Cited By ~ 7

Author(s):

PETR VOPĚNKA

Keyword(s):

Set Theory ◽

Philosophical Foundations ◽

Alternative Set

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Naturalness and Artificiality Revisited Through Natural Infinity

FILOSOFIE DNES ◽

10.26806/fd.v11i2.277 ◽

2020 ◽

Vol 11 (2) ◽

Author(s):

Jan Romportl

Keyword(s):

Set Theory ◽

Cognitive Enhancement ◽

Formal Model ◽

Structural Difference ◽

Human Personality ◽

Alternative Set

Discussions about naturalness, artificiality and unnaturalness in this article are motivated by the field of Human Cognitive Enhancement (HCE) because of its potential for altering human personality and identity. This article at first proposes a concept of human naturalness as interaction between physis and logos. Then it presents an intuitive understanding of naturalness in terms of the inherent inability of language to fully describe all attributes of an object that is natural. The analytical core of the article proposes a formal model of naturalness utilising Vopenka's phenomenologically grounded Alternative Set Theory (AST), comprising and formalising the concept of natural infinity. A brief introduction to AST is presented as well. Naturalness and artificiality are modelled as two structurally different naturally infinite semisets within AST. This key structural difference is then analysed and applied back to the domain of HCE.

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