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A.P. Morse’s Set Theory and AnalysisSet Theory

A.P. Morse’s Set Theory and Analysis: Set Theory Chapter 2. Set Theory Any set theory formalized in the spirit of Chapter 0 naturally suggests certain simple provocative questions. When is x? What is (x ∈ y)? What is (p→ q)? What is x ux? We shall commit ourselves to a set theory unorthodox in the sense that these questions are answered. In fact, our Axioms 2.5.0-2.5.3 were conceived for this express purpose. We were led to these four axioms somewhat as follows. We believe every (mathematical) thing is a set. We believe there is no di erence between the conjunction of two or more things and their intersection. We believe there is no di erence between the disjunction of two things and their union. We believe there is no di erence between the negation of a thing and its complement. We have come to believe a thing if and only if the empty set is a member of the thing. We believe (x ∈ y) if and only if x is a member of y. We believe (x ∈ y) if and only if (x ∈ y) is the universe. We disbelieve (x ∈ y) if and only if (x ∈ y) is the empty set. PRELIMINARIES http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A.P. Morse’s Set Theory and AnalysisSet Theory

Editors: Alps, Robert A.

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Publisher
Springer International Publishing
Copyright
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
ISBN
978-3-031-05354-2
Pages
41 –105
DOI
10.1007/978-3-031-05355-9_3
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 2. Set Theory Any set theory formalized in the spirit of Chapter 0 naturally suggests certain simple provocative questions. When is x? What is (x ∈ y)? What is (p→ q)? What is x ux? We shall commit ourselves to a set theory unorthodox in the sense that these questions are answered. In fact, our Axioms 2.5.0-2.5.3 were conceived for this express purpose. We were led to these four axioms somewhat as follows. We believe every (mathematical) thing is a set. We believe there is no di erence between the conjunction of two or more things and their intersection. We believe there is no di erence between the disjunction of two things and their union. We believe there is no di erence between the negation of a thing and its complement. We have come to believe a thing if and only if the empty set is a member of the thing. We believe (x ∈ y) if and only if x is a member of y. We believe (x ∈ y) if and only if (x ∈ y) is the universe. We disbelieve (x ∈ y) if and only if (x ∈ y) is the empty set. PRELIMINARIES

Published: Aug 25, 2022

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