[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 73 (1002ﾚｽ)
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672(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/08/16(土) 07:30:11.74 ID:psDSFTci(2/9) AAS
つづき
Weaker systems
Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice.[6] However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice.
For example, Vω∖{∅} has a choice function, where Vω is the set of hereditarily finite sets, i.e. the first set in the Von Neumann universe of non-finite rank.
The choice function is (trivially) the least element in the well-ordering.
Another example is the set of proper and bounded open intervals of real numbers with rational endpoints.
ZF+ACω suffices to prove that the union of countably many countable sets is countable. These statements are not equivalent: Cohen's First Model supplies an example where countable unions of countable sets are countable, but where ACω does not hold.[7]
https://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Axiom of dependent choice
In mathematics, the axiom of dependent choice, denoted by
DC, is a weak form of the axiom of choice (AC) that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article in reverse mathematics that explores which set-theoretic axioms are needed to develop analysis.[a]
Relation with other axioms
Unlike full AC, DC is insufficient to prove (given ZF) that there is a non-measurable set of real numbers, or that there is a set of real numbers without the property of Baire or without the perfect set property. This follows because the Solovay model satisfies ZF+DC, and every set of real numbers in this model is Lebesgue measurable, has the Baire property and has the perfect set property.
The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[4][5]
It is possible to generalize the axiom to produce transfinite sequences.
If these are allowed to be arbitrarily long, then it becomes equivalent to the full axiom of choice.
https://ja.wikipedia.org/wiki/%E5%BE%93%E5%B1%9E%E9%81%B8%E6%8A%9E%E5%85%AC%E7%90%86
従属選択公理(英語: axiom of dependent choice; DCと略される)
(引用終り)
以上
865(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/08/20(水) 16:34:32.47 ID:n7uBTsIt(4/5) AAS
>>681
>整列可能定理の証明の方法で可算集合Xの整列順序を作るには選択関数f:2^X-{}→Xが必要。且つ|2^X-{}|は非可算。よって可算選択公理は役に立たない。
>一方で全単射g:N→Xが存在するからg(0)<g(1)<・・・で整列順序<を定義可能。（よって整列可能定理の証明の方法を取る必要が無い。よっていかなるタイプの選択公理も不要。）
中高一貫生も来る可能性があるので、赤ペン先生をしておく
まず
(参考)>>671-672より再録
１）下記 可算選択公理 Axiom of countable choice ACω は
”Application of ACω yields a sequence (Bn) n∈N ”
つまり ω長さの sequence (Bn) n∈N を作る能力がある
２）一方 Axiom of dependent choice DC は 下記
”The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[4][5]
It is possible to generalize the axiom to produce transfinite sequences.
If these are allowed to be arbitrarily long, then it becomes equivalent to the full axiom of choice.”
３）要するに、DC は ACωより強力で ωを超えて ”produce transfinite sequences”だ
また ”If these are allowed to be arbitrarily long, then it becomes equivalent to the full axiom of choice.”
ってこと。つまりは、種々の選択公理の能力は、生成できる列長さで 測ることができる■
https://en.wikipedia.org/wiki/Axiom_of_countable_choice
Axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function.
Applications
ACω is particularly useful for the development of mathematical analysis, where many results depend on having a choice function for a countable collection of sets of real numbers.
Example: infinite implies Dedekind-infinite
As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite:
Let X be infinite. For each natural number n, let An be the set of all n-tuples of distinct elements of X.
Since X is infinite, each An is non-empty.
Application of ACω yields a sequence (Bn) n∈N where each Bn is an n-tuple.
One can then concatenate these tuples into a single sequence (bn)n∈N of elements of X, possibly with repeating elements.
Weaker systems
Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice. However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice.
For example, Vω∖{∅} has a choice function, where Vω is the set of hereditarily finite sets, i.e. the first set in the Von Neumann universe of non-finite rank.
The choice function is (trivially) the least element in the well-ordering.
つづく
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