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| NEW AXIOM SYSTEMS FOR THE SET THEORY |
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Citation： |
Mo Shaokui.NEW AXIOM SYSTEMS FOR THE SET THEORY[J].Chinese Annals of Mathematics B,1980,1(2):309~316 |
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Net amount： 853 |
Authors： |
Mo Shaokui; |
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| Abstract： |
In the present paper we give four axiom systems for the set theory.
First,the ZFC system may be given by:(1) the axiom of extensionality,(5) the axiom of replacement,(6) teh axiom of infinity,(7) teh axiom of regularity,(8) the axiom of choice and (A) the strong pairing set axiom, i.e.,\(\exists y\forall x(x\varepsilon y \equiv x \le a \vee x \le b)\), where \(x \le a\) will be defined in the next.
Second, the ZFC system may also be given by (1)(6)(7)(8), and (B) the strong axiom of replacement, i.e., Fcn \(\phi \supset \exists y\forall x(x\varepsilon y \equiv \exists t{\kern 1pt} (t \le a \vee t \le b.{\kern 1pt} {\kern 1pt} \wedge \phi (t,x)))\), where \(\phi \) means that \(\phi \) is a function,i.e., \(\forall x\forall y\forall z(\phi (x,y) \wedge \phi (t,x)) \cdot \supset y = z)\).
Third, the ZFC system may be strenthened as follows, (1)(6)(7)(A) and (C) the first strong axiom of concretion,i.e., \(\exists y\forall x(x\varepsilon y \equiv \psi (x)) \equiv \nabla (\psi )\) where \(\nabla (\psi )\) means that \(\forall \varphi {\kern 1pt} {\kern 1pt} {\kern 1pt} (Fcn{\kern 1pt} {\kern 1pt} {\kern 1pt} \varphi \supset \exists u\bar \exists x(\psi (x) \wedge \psi (x,u)))\).
Fourth,the ZFC system may also be strengthened as follows,(1)(2) the pairing set axiom, (3) the union set axiom, (4) the power set axiom, (6)(7) and (D) the second strong axiom of concretion, i.e.,\(\exists y\forall x(x\varepsilon y \equiv \psi (x)) \equiv \cdot \nabla (\psi ) \vee \nabla (\bar \psi )\).
We should note that the last system possesses a kind of symmetry such that in it we may have the universal set, the complementary operation, and hence the principle of duality. |
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