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 3. A ⊆ A. PROOF. Since it is a truth of logic that (∀x)(x ∈ A → x ∈ A), it follows immediately from Definition 3 that Theorem 3 asserts simply that inclusion is reflexive; the next theorem asserts that it has the property ...
... prove. THEOREM 12. x ∈ A ∩ B ↔ x ∈ A & x ∈ B. PROOF. Using the identity: A ∩ B = A ∩ B and putting 'A ∩ B' for 'y' in Definition 5, we obtain the theorem at once. Q.E.D. The next two theorems assert the commutativity and ...
... Theorem 2 A ∩ 0 = 0. Q.E.D THEOREM 17. A ∩ B ⊆ A. THEOREM 18. A ⊆ B ↔ A ∩ B = A. We now turn to the theorem justifying the operation of union of sets. The proof of this theorem involves the first use of the union axiom. THEOREM 19 ...
... 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 B. THEOREM 26. A G B — A U B = B. THEOREM 27. A C C & B C C → A U B C C. We ... proof of Theorem 28 a device which is.
Patrick Suppes. In the proof of Theorem 28 a device which is used over and over again has been employed: in order to ... THEOREM 29. (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C). We next state the justifying theorem and definition of the ...
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 | |