On the extension principle in internal set theory

1992 ◽  
Vol 33 (6) ◽  
pp. 999-1010
Author(s):  
V. G. Kanoveį
Keyword(s):  
Set Theory ◽  
Extension Principle ◽  
Internal Set

scholarly journals Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2048
Author(s):  
Ileana Ruxandra Badea ◽  
Carmen Elena Mocanu ◽  
Florin F. Nichita ◽  
Ovidiu Păsărescu
Keyword(s):  
Mathematical Modeling ◽  
Artificial Intelligence ◽  
Mathematical Model ◽  
Set Theory ◽  
Standard Analysis ◽  
New Methods ◽  
Internal Set ◽  
Human Thinking ◽  
Global Brain ◽  
Non Standard Analysis

The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.

scholarly journals Some properties of integration of real-valued function over a fuzzy interval

2021 ◽  
Vol 8 (12) ◽  
pp. 9-13
Author(s):  
M. A. Shakhatreh ◽  
◽  
A. M. Al-Shorman ◽  
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Interval Analysis ◽  
Fuzzy Set Theory ◽  
Fuzzy Numbers ◽  
Extension Principle ◽  
Mathematical Concepts ◽  
Fuzzy Distance ◽  
Real Algebra ◽  
Fuzzy Interval

One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many structural qualities. The extension concept is demonstrated to be particularly beneficial for defining set-theoretic operations for higher fuzzy sets. We need some definitions related to our properties before we can create the properties of integration of a crisp real-valued function over a fuzzy interval. It is our goal in this article to develop and demonstrate certain characteristics of a real-valued function over a fuzzy interval in order to broaden the scope of the notion of integration of a real-valued function over a fuzzy interval. Some of these characteristics are linked to the operations of extended addition and extended subtraction, while others are not.

Uniqueness, collection, and external collapse of cardinals in IST and models of Peano arithmetic

1995 ◽  
Vol 60 (1) ◽  
pp. 318-324 ◽  
Author(s):  
V. Kanovei
Keyword(s):  
Set Theory ◽  
Peano Arithmetic ◽  
General Technique ◽  
Nonstandard Model ◽  
Internal Set

AbstractWe prove that in IST, Nelson's internal set theory, the Uniqueness and Collection principles, hold for all (including external) formulas. A corollary of the Collection theorem shows that in IST there are no definable mappings of a set X onto a set Y of greater (not equal) cardinality unless both sets are finite and #(Y) ≤ n #(X) for some standard n. Proofs are based on a rather general technique which may be applied to other nonstandard structures. In particular we prove that in a nonstandard model of PA, Peano arithmetic, every hyperinteger uniquely definable by a formula of the PA language extended by the predicate of standardness, can be defined also by a pure PA formula.

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