[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
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541(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/30(土)20:55 ID:4Ujjq2jv(6/17) AAS
>>540
つづき

?As we know, the set of all objects (if it exists) enjoys paradoxical properties: unlike a theorem known to G. Cantor, the power of this set would not be inferior to that of the class of all its subassemblies.
It is the same of the class composed of all the sets
containing a single element; therefore, K classes do not check, Cantor's theorem.
?Taking this fact into account, one could question the very existence of classes K.

By modifying Mr. Sierpinski's definition so as to remove that drawback, I get the following definition:
省6

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542(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/30(土)20:55 ID:4Ujjq2jv(7/17) AAS
>>541
つづき

?We can show that a finite set according to this definition is also in the ordinary sense and reciprocally.
In other words: for a set to be finite according to the proposed definition, it is necessary and sufficient that the number of its elements can be expressed by a natural number (the notion of natural number being assumed to be known).
?Indeed, let M be a set whose number of elements can be expressed by a natural number; let Z be any class satisfying the conditions 1-3.
We will show that every subset of M belongs to Z.
This is - under condition 2 - subsets composed of a single element; at the same time, if this is so subsets containing n elements, it is the same - according to 3 - of those which contain n + 1.
省8

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