>>242 (引用開始) 3）とすると、(assuming countable choice) ならば、>>239より ”1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x,” だから、不足しているのは Rが ”Metric” であることだが。”Rが Metric”をいうには、countable choice だけでは 不足なのかな？ (引用終り)
下記 Construction of the real numbers の Construction from Cauchy sequences で metric spaces として completion（完備）までやっているが、どの選択公理を使うかの記述がない ”axiom of dependent choice”だと思うのだが・・ （＾＾
(参考) https://en.wikipedia.org/wiki/Construction_of_the_real_numbers Construction of the real numbers Explicit constructions of models
Construction from Cauchy sequences A standard procedure to force all Cauchy sequences in a metric space to converge is adding new points to the metric space in a process called completion. R is defined as the completion of the set Q of the rational numbers with respect to the metric |x － y| Normally, metrics are defined with real numbers as values, but this does not make the construction/definition circular, since all numbers that are implied (even implicitly) are rational numbers.[5]
An advantage of constructing R as the completion of Q is that this construction can be used for every other metric spaces.
つまり、整列可能定理は公理として、有理コーシー列で有理数Qの完備化を可能として ↓ つまり、整列可能定理は公理として、x∈R subset A⊂R で 有理コーシー列 a sequence in A＼{x} that converges to x で有理数Qの完備化を可能として(但し、RをcompactにするためDCを使用>>261)
(参考) >>236より下記（Equivalent are:1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x, ＆ 9. the Axiom of Choice for countable collections of subsets of R.） archive.wikiwix.com/cache/display2.php?url=http%3A%2F%2Fwww.emis.de%2Fjournals%2FCMUC%2Fpdf%2Fcmuc9703%2Fherrli.pdf Comment.Math.Univ.Carolin. 38,3(1997)545–552 545 Choice principles in elementary topology and analysis Horst Herrlich 1. In the realm of the reals We start by observing that several familiar topological properties of the reals are equivalent to each other and to rather natural choice-principles. Theorem 1.1 ([15], [29], [30]). Equivalent are: 1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x, 2. a function f : R → R is continuous at a point x iff it is sequentially continuous at x, 3. a real-valued function f : A → R from a subspace A of R is continuous iff it is sequentially continuous, 4. each subspace of R is separable, 5. R is a Lindel¨ of space, 6. Q is a Lindel¨ of space, 7. N is a Lindel¨ of space, 8. each unbounded subset of R contains an unbounded sequence, 9. the Axiom of Choice for countable collections of subsets of R. There exist models of ZF that violate the above conditions ([17], [18]). Observe the fine distinction between conditions 2 and 3 of Theorem 1.1. These may lead one to assume that also the following property is equivalent to the above conditions: (*) a function f : R －→ R is continuous iff it is sequentially continuous. However, this would be a serious mistake: (*) holds in ZF (without any choiceassumptions) — see [29]. If, however, we consider functions f : X －→ R with metric domain we need even more choice than in Theorem 1.1, — see Theorem 2.1.
赤ペン先生、入ります！ｗｗ ；ｐ） 「なんで整列定理が必要と思ったの？」については、下記のHorst Herrlichの ”Theorem 2.4 ([4], [14]). Equivalent are: 1. in a (pseudo)metric space X, a point x is an accumulation point of a subset A iff there exists a sequence in A\ {x} that converges to x, 略す 17. the Axiom of Countable Choice.” を、百回音読してね ；ｐ）
なお、下記のソロヴェイモデル 到達不能基数＋ ”ZF + DC を満たしで 実数集合が全てルベーグ可測で perfect set property を持ち、ベールの性質を持つものになっている” ここの部分は、到達不能基数が ZFCの外です だから、到達不能基数＋”ZF + DC と、”17. the Axiom of Countable Choice”は、直ちには矛盾していないことを付言しておきます ；ｐ） （本音は、良く分からないｗ）
(参考) 下記（Equivalent are:1. in R, a point x is an accumulation point of a subset A iff there exists a sequence in A＼{x} that converges to x, ＆ 9. the Axiom of Choice for countable collections of subsets of R.） archive.wikiwix.com/cache/display2.php?url=http%3A%2F%2Fwww.emis.de%2Fjournals%2FCMUC%2Fpdf%2Fcmuc9703%2Fherrli.pdf Comment.Math.Univ.Carolin. 38,3(1997)545–552 545 Choice principles in elementary topology and analysis Horst Herrlich P546 2. In the realm of pseudometric spaces In this section we consider (pseudo)metric spaces and various compactness-notions for them.
Theorem 2.1 ([4], [15]). Equivalent are: 1. every separable pseudometric space is a Lindel¨ of space, 2. every pseudometric space with a countable base is a Lindel¨ of space, 3. the Axiom of Choice for countable collections of subsets of R.
Definition 2.2. A pseudometric space X is called 1. Heine-Borel-compact provided every open cover of X contains a finite one, 2. Weierstraß-compact provided for every infinite subset of X there exists an accumulation point, 3. Alexandroff-Urysohn-compact provided for every infinite subset of X there exists a complete accumulation point, 4. sequentially-compact provided every sequence in X has a convergent subsequence. Under the Axiom of Choice the above compactness concepts are equivalent. This is no longer the case in ZF.
Theorem 2.4 ([4], [14]). Equivalent are: 1. in a (pseudo)metric space X, a point x is an accumulation point of a subset A iff there exists a sequence in A\ {x} that converges to x, 略す 17. the Axiom of Countable Choice. The Axiom of Dependent Choices implies the Baire Category Theorem for complete pseudometric spaces, and the latter implies the Axiom of Countable Choice.
(参考) ja.wikipedia.org/wiki/%E3%82%BD%E3%83%AD%E3%83%B4%E3%82%A7%E3%82%A4%E3%83%A2%E3%83%87%E3%83%AB ソロヴェイモデルはロバート M. ソロヴェイ (1970)によって構成されたモデルでツェルメロ＝フレンケル集合論 (ZF) の全ての公理が成り立ち、選択公理を除去し、実数の集合が全てルベーグ可測であるようにしたものである。この構成は到達不能基数の存在に依拠している。 構成 ソロヴェイはそのモデルを二つのステップによって構成した。まず初めに、到達不能基数 κ を含む ZFC のモデル M から始める。 略す 二つ目のステップではソロヴェイのモデル N として、M[G] の中で順序数の可算列で遺伝的に定義可能な集合全てからなるクラスを考える。このモデル N は M[G] の内部モデルであって ZF + DC を満たし、実数集合が全てルベーグ可測で perfect set property を持ち、ベールの性質を持つものになっている。この証明には、M[G] の実数は全て順序数の可算列を用いて定義可能であり、N と M[G] が同じ実数を持っていることを使う。 略す (引用終り) 以上
(参考) archive.wikiwix.com/cache/display2.php?url=http%3A%2F%2Fwww.emis.de%2Fjournals%2FCMUC%2Fpdf%2Fcmuc9703%2Fherrli.pdf Comment.Math.Univ.Carolin. 38,3(1997)545–552 545 Choice principles in elementary topology and analysis Horst Herrlich P546 2. In the realm of pseudometric spaces In this section we consider (pseudo)metric spaces and various compactness-notions for them.
Definition 2.2. A pseudometric space X is called 1. Heine-Borel-compact provided every open cover of X contains a finite one, 2. Weierstraß-compact provided for every infinite subset of X there exists an accumulation point, 3. Alexandroff-Urysohn-compact provided for every infinite subset of X there exists a complete accumulation point, 4. sequentially-compact provided every sequence in X has a convergent subsequence. Under the Axiom of Choice the above compactness concepts are equivalent. This is no longer the case in ZF.
Theorem 2.4 ([4], [14]). Equivalent are: 1. in a (pseudo)metric space X, a point x is an accumulation point of a subset A iff there exists a sequence in A＼ {x} that converges to x, 略す 17. the Axiom of Countable Choice. The Axiom of Dependent Choices implies the Baire Category Theorem for complete pseudometric spaces, and the latter implies the Axiom of Countable Choice.
>>270 >Choice principles in elementary topology and analysis Horst Herrlich
Horst Herrlichは、下記か 大物ですな （＾＾
(参考) en.wikipedia.org/wiki/Horst_Herrlich Horst Herrlich (11 September 1937, in Berlin – 13
304 名前： March 2015, in Bremen) was a German mathematician, known as a pioneer of categorical topology.
Education and career From 1971 to 2002 Herrlich was a professor of mathematics with a focus on general topology and category theory at the University of Bremen.
He was an Invited Speaker of the International Congress of Mathematicians in 1974 in Vancouver.[4] He is regarded as a founder of categorical topology, which deals with general topology using the methods of category theory.
books.google.co.jp/books?id=_0cDCAAAQBAJ&redir_esc=y Axiom of Choice 前表紙 Horst Herrlich Springer, 2006/07/21 - 198 ページ AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that:
- Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC).
- Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games).
- Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness).
Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory. []
下記”The axiom of choice in metric measure spaces and maximal δ-separated sets” ”可算選択公理は、擬距離空間上のボレル測度の存在が、開球の測度が正で有限であることから、その空間の可分性を意味することを証明するのに必要かつ十分であることを示す” ボレル測度や、ルベーグ測度を作るのに、ZFCが必要か はたまた ZF＋DC(従属選択)でよいのか？ それが問題だ by ハムレット ；ｐ） 調査中
(参考) link.springer.com/article/10.1007/s00153-023-00868-4 Archive for Mathematical Logic The axiom of choice in metric measure spaces and maximal δ-separated sets Michał Dybowski & Przemysław Górka Volume 62 (2023) Abstract We show that the Axiom of Countable Choice is necessary and sufficient to prove that the existence of a Borel measure on a pseudometric space such that the measure of open balls is positive and finite implies separability of the space. In this way a negative answer to an open problem formulated in Górka (Am Math Mon 128:84–86, 2020) is given. Moreover, we study existence of maximal δ-separated sets in metric and pseudometric spaces from the point of view the Axiom of Choice and its weaker forms. (google訳) 可算選択公理は、擬距離空間上のボレル測度の存在が、開球の測度が正で有限であることから、その空間の可分性を意味することを証明するのに必要かつ十分であることを示す このようにして、Górka (Am Math Mon 128:84–86, 2020) で定式化された未解決問題に対する否定的な答えが与えられる
www.sciencedirect.com/science/article/pii/S0166864197000400?via%3Dihub Topology and its Applications Volume 82, Issues 1–3, 23 January 1998, Pages 3-14 T. Hoshina 、J.
Proof from axiom of choice The well-ordering theorem follows from the axiom of choice as follows.[9]
Let the set we are trying to well-order be A, and let f be a choice function for the family of non-empty subsets of A. For every ordinal α, define an element aα that is in A by setting aα= f(A∖{aξ∣ξ<α}) if this complement A∖{aξ∣ξ<α} is nonempty, or leave aα undefined if it is. That is, aα is chosen from the set of elements of A that have not yet been assigned a place in the ordering (or undefined if the entirety of A has been successfully enumerated). Then the order < on A defined by aα<aβ if and only if α<β (in the usual well-order of the ordinals) is a well-order of A as desired, of order type sup{α∣aα is defined}.
>> 「A∖{aξ∣ξ<α} が空となれば完結する、ということだと思うけど >> そのようなξが存在する、という保証は？」 > wikipedia の証明の最後 > ”a well-order of A as desired, of order type sup{α∣aα is defined}.” > が、”そのようなξが存在する、という保証”だね
それ、肝心の sup{α∣aα is defined} の存在を保証してないけど 英語読めない？ それとも日本語に翻訳してもそもそも文章が読めない？ 前者なら、英語勉強して 後者なら、国語勉強して
集合の濃度と基数 →詳細は「濃度 (数学)」を参照 集合 A から集合 B への全単射が存在するとき、A と B は同数 (equinumerous) であるといい、A ≈ B で表す。 選択公理を仮定すれば、整列定理により任意の集合 A に対して A と同数であるような順序数が存在することが言える。***) そこで、集合 A と同数であるような順序数の中で最小のものを A の濃度 (cardinality of A) といい、これを |A| あるいは card(A) で表す。ある集合 A に対して α = |A| である順序数 α を基数 (cardinal number) と呼ぶ。集合の濃度に関して次が成り立つ： |A| = |B| ⇔ A ≈ B A が有限集合のとき、|A| は A の要素の個数に等しい。 基数に対しても、上で定義した順序数の演算とは別に和、積、冪を定義することができる。
