A.P. Morse’s Set Theory and Analysis: Set Theory
Alps, Robert A.
2022-08-25 00:00:00
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
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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
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