Axiomatic Set TheoryCourier Corporation, 4 mai 2012 - 265 pages One of the most pressingproblems of mathematics over the last hundred years has been the question: What is a number? One of the most impressive answers has been the axiomatic development of set theory. The question raised is: "Exactly what assumptions, beyond those of elementary logic, are required as a basis for modern mathematics?" Answering this question by means of the Zermelo-Fraenkel system, Professor Suppes' coverage is the best treatment of axiomatic set theory for the mathematics student on the upper undergraduate or graduate level. |
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... theorem asserts the idempotence of intersection. THEOREM 15. A ∩ A = A. PROOF. By Theorem 12 x ∈ A ∩ A ↔ x ∈ A & x ∈ A, but x ∈ A & x ∈ A ↔ x ∈ A, and thus by Definition 5 A ∩ A = A. Q.E.D Three intuitively obvious theorems ...
... theorem for the operation of union analogous to Theorem 12. THEOREM 20. x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B. PROOF. Similar to the proof of Theorem 12. Since many of the proofs of theorems about union of sets parallel those about ...
... THEOREM 23. A U A = A. Further facts are asserted in the next four theorems. THEOREM 24. A U A = A. THEOREM 25. A C A U ... 12 x e (A U B) s C ex e A U B & x e C, and by Theorem 20 x e A U B & x e C – (x e A V x e B) & x e C, and by the ...
... Theorem 11, but here taking φ(x) as 'x ∉ B'. DEFINITION 7. A ~ B = y ↔ (∀x)(x ∈ y ↔ x ∈ A & x ∉ B) & y is a set. THEOREM 31. x ∈ A ~ B ↔ x ∈ A & x ∉ B. PROOF. Similar to the proof of Theorem 12. The next theorem obviously ...
... theorem proceeds by the line of argument of Russell's paradox via a reductio ad absurdum. THEOREM 41. −(∃A)(∀x)(x ∈ A ... 12. We define the operation of symmetric difference by the identity: 1. 2. 3. 4. A ÷ B = (A ~ B) ∪ (B ~A).
Table des matières
RELATIONS AND FUNCTIONS | |
EQUIPOLLENCE FINITE SETS AND CARDINAL NUMBERS | |
FINITE ORDINALS AND DENUMERABLE SETS | |
RATIONAL NUMBERS AND REAL NUMBERS | |
TRANSFINITE INDUCTION AND ORDINAL ARITHMETIC | |
THE AXIOM OF CHOICE | |
REFERENCES | |
AUTHOR INDEX | |