[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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91(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/05(土)21:31 ID:JrhjRl4x(40/46) AAS
>>77
ツェルメロ構成
批判はされているけれど（＾＾
https://plato.stanford.edu/entries/zermelo-set-theory/
Stanford Encyclopedia of Philosophy
Zermelo’s Axiomatization of Set Theory
First published Tue Jul 2, 2013
(抜粋)
3.2.1 Representing Ordinary Mathematics
The first obvious question concerns the representation of the ordinary number systems.
The natural numbers are represented by Zermelo as by Φ, {Φ}, {{Φ}}, …, and the Axiom of Infinity gives us a set of these.
Moreover, it seems that, since both the set of natural numbers and the power set axiom are available, there are enough sets to represent the rationals and the reals, functions on reals etc.
What are missing, though, are the details: how exactly does one represent the right equivalence classes, sequences etc.?
And assuming that one could define the real numbers, how does one characterise the field operations on them?
In addition, as mentioned previously, Zermelo has no natural way of representing either the general notions of relation or of function.
This means that his presentation of set theory has no natural way of representing those parts of mathematics (like real analysis) in which the general notion of function plays a fundamental part.
3.2.2 Ordinality
Zermelo's idea (1908a) was pursued by Kuratowski in the 1920s, thereby generalising and systematising work, not just of Zermelo, but of Hessenberg and Hausdorff too, giving a simple set of necessary and sufficient conditions for a subset ordering to represent a linear ordering.
He also argues forcefully that it is in fact undesirable for set theory to go beyond this and present a general theory of ordinal numbers:
(引用終り)
92(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/10/05(土)21:35 ID:JrhjRl4x(41/46) AAS
>>91 補足
”The natural numbers are represented by Zermelo as by Φ, {Φ}, {{Φ}}, …, and the Axiom of Infinity gives us a set of these.
Moreover, it seems that, since both the set of natural numbers and the power set axiom are available, there are enough sets to represent the rationals and the reals, functions on reals etc.
What are missing, though, are the details: how exactly does one represent the right equivalence classes, sequences etc.?”
ツェルメロ自然数構成
批判はされているけれど（＾＾
・by Φ, {Φ}, {{Φ}}, …, and the Axiom of Infinity gives us a set of these
・since both the set of natural numbers and the power set axiom are available, there are enough sets to represent the rationals and the reals, functions on reals etc.
・何が不足なの？ What are missing, though, are the details: how exactly does one represent the right equivalence classes, sequences etc.?
まあ、ツェルメロ自然数構成から、無限集合が出来て、自然数とその冪集合から、有理数や実数や実関数などはできる
でも、批判はあった。それは、基礎論パイオニアの宿命でもあったかもしれない（＾＾
95(3): 2019/10/05(土)21:51 ID:kZwmbLNI(39/44) AAS
>>91-92
英語読めませんか？
Infinity
This final axiom asserts the existence of an infinitely large set which contains the empty set, and for each set a that it contains, also contains the set {a}. (Thus, this infinite set must contain ∅, {∅}, {{∅}}, ….)
つまり>>29で述べたω’(＝{{},{{}},{{{}}},…})
∃ω’.{}∈ω’∧(∀x.x∈ω’⇒{x}∈ω’)
だといってます
決して{・・・{Φ}・・・}ではありません
501(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/27(水)21:17 ID:qnEhNItW(6/12) AAS
>>491 補足
すでに、このスレの>>91に示したように、
天才Zermeloが、シングルトンによる自然数の構成を与えた（1908年）
（”The natural numbers are represented by Zermelo as by Φ, {Φ}, {{Φ}}, …, and the Axiom of Infinity gives us a set of these.”）
そして、確かに、Zermeloの構成は批判され、その後ノイマン構成が採用された
だが、天才Zermeloのシングルトンによる自然数の構成が決して間違っていた訳では無い
無数の超準モデルの1つだよ。（レーヴェンハイム＝スコーレムの定理）
そのことに無知な、落ちこぼれたちｗｗ（＾＾；
（>>91より再録）
https://plato.stanford.edu/entries/zermelo-set-theory/
Stanford Encyclopedia of Philosophy
Zermelo’s Axiomatization of Set Theory
First published Tue Jul 2, 2013
(抜粋)
3.2.1 Representing Ordinary Mathematics
省3
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