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Safe Set Theory
This chapter introduces Russell’s idea that sets are built by stages. This avoids the obvious contradictions of the naïve set theory of Cantor’s that...
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Set Theory
In this chapter, I present the foundational results of set theory. I introduce all the axioms for the set theory and then build up the necessary...
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Some Elementary Informal Set Theory
Set theory is due to Georg Cantor. “Elementary” in the title above does not apply to the body of his work, since he went into considerable technical...
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Intuitionistic sets and numbers: small set theory and Heyting arithmetic
It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set...
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Probability Calibration with Fuzzy Set Theory to Improve Early Cancer Detection
AbstractCancer is the leading cause of death before the age of 70 years. An important step for reducing the cancer mortality can be its early...
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Chain-dependent Conditions in Extremal Set Theory
In extremal set theory our usual goal is to find the maximal size of a family of subsets of an n -element set satisfying a condition. A condition is...
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A heuristic approximation algorithm of minimum dominating set based on rough set theory
The minimum dominating set of graph has been widely used in many fields, but its solution is NP-hard. The complexity and approximation accuracy of...
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Set Theory
Modern analysis, like most other subfields of modern mathematics, is concerned with numbers, sets, and geometry. We have already introduced one type...
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Conley index theory without index pairs. I: The point-set level theory
We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce,...
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Incompleteness of Arithmetic from the Viewpoint of Diophantine Set Theory
The authors analyze Diophantine sets and show that all recursively enumerable sets are Diophantine. Based on the classical results from the theory of...
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Effective neutrosophic soft set theory and its application to decision-making
The neutrosophic soft set is the most powerful and effective extension of soft sets, which deals with parameterized values of options since it also...
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Return to Set Theory
This chapter introduces the second part of the book. Why should we bother about set theory, again? And what is axiomatic set theory? In some sense,...
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A few more dissimilarities between second-order arithmetic and set theory
Second-order arithmetic and class theory are second-order theories of mathematical subjects of foundational importance, namely, arithmetic and set...
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A Universal Algebraic Set Theory Built on Mereology with Applications
Category theory is often treated as an algebraic foundation for mathematics, and the widely known algebraization of ZF set theory in terms of this...
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The Convex-Set Algebra and Intersection Theory on the Toric Riemann-Zariski Space
In previous work of the author, a top intersection product of toric b -divisors on a smooth complete toric variety is defined. It is shown that a nef...
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Decision analysis review on the concept of class for bipolar soft set theory
To eliminate uncertainty, all data expressed by decision-makers must be processed correctly. The most useful mathematical models developed for this...
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Second order arithmetic as the model companion of set theory
This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model...