現代数学の系譜　カントル　超限集合論

現代数学の系譜　カントル　超限集合論at MATH

現代数学の系譜　カントル　超限集合論 - 暇つぶし2ch678: M = E N and N = E M at the same time, then M = N is always M or shorter: every set is determined by its elements. P266 But in order to secure the existence of "infinite" sets, we still need the following axiom, which derives from its essential content by Mr. R. Dedekind. Axiom VII. The domain contains at least a set Z which contains the null set as an element and is such that each of its elements a is another element of the form {a}, or which with each of its elements a is also the corresponding set {a } as an element. (Axiom of the infinite.) 14 VII. *) If Z is an arbitrary set of the properties required in VII, then for each of its subsets Z1 it is definite whether it possesses the same property. For if a is any element of Z1 ', it is definite whether {a} ∈ Z1, and all the elements a of Z1 thus constituted form the elements of a subset Z1' for which it is definite whether Z1 '= Z1 or Not. Thus, all subsets Z1 of the considered property form the elements of a subset T = E UZ, and the average corresponding to them (# 9) Z0 = DT is an amount of the same nature. つづく

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