＜Lindelöfとは？＞ en.wikipedia.org/wiki/Lindel%C3%B6f_space Lindelöf space In mathematics, a Lindelöf space[1][2] is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.
（注：上記の”(*) a function f : R －→ R is continuous iff it is sequentially continuous. (*) holds in ZF (without any choiceassumptions) — see [29]”が、下記と思う） alg-d.com/math/ac/continuous.html トップ > 数学 > 選択公理 > 実数関数の連続性 壱大整域 20130323 一方，次の命題はZFで証明できる． 命題 f: R→Rとする． fがRで連続 ⇔ 収束点列 { xn }n=0∞に対して limn→∞f(xn) = f(limn→∞xn) 証明 略す
ja.wikipedia.org/wiki/%E5%AE%9F%E6%95%B0%E3%81%AE%E9%80%A3%E7%B6%9A%E6%80%A7 実数の連続性（continuity of real numbers）とは、実数の集合がもつ性質である。有理数はこの性質を持たない。 実数の連続性は、実数の完備性 (completeness of the real numbers) とも言われる また、実数の連続性を議論の前提とする立場であれば実数の公理と記述する場合もある。 なお、ここで言う連続性は、関数の連続性とは別の概念である。 実数の連続性と同値な命題 実数の連続性と同値な命題は多数存在する。順序体（位相は順序位相を入れる）において、実数の公理は
en.wikipedia.org/wiki/Compact_space Compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.[1] The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers Q is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers R is not compact either, because it excludes the two limiting values +∞ and －∞. However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.
＜注：下記は、対角線論法でない 実数Ｒの非可算の証明の話＞ en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article Cantor's first set theory article This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article was published in 1874. In 1879, he modified his uncountability proof by using the topological notion of a set being dense in an interval.
＜付録＞ これ面白いね Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a "Tarski universe" it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than ZFC. For example, adding this axiom supports category theory. The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs. (引用終り) 以上
可算選択公理でさえ、R is a Lindel や 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, ↓ 可算選択公理でさえ、R is a Lindelöf や 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,
(参考) https://mathoverflow.net/questions/218874/some-axiom-of-choice-and-dependent-choice-issues mathoverflow Some "axiom of choice" and "dependent choice" issues asked Sep 21, 2015 Julian Newman
I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these.
If I understand correctly, mathematicians tend to be quite happy working with ZF+DC, but other forms of choice that are not implied by DC can be more controversial.
[Therefore it seems natural that people should give higher priority to discussing the differences in provable theorems between ZFC and ZF+DC -- or at least, the differences in provable theorems between ZFC and ZF+(countable choice) -- than to discussing the differences in provable theorems between ZFC and ZF. (Indeed, you basically can't do any analysis in just ZF.)]
My questions are:
Is it consistent with ZF+DC that every subset of R is Borel-measurable? If the answer to Q1 is no: Is it consistent with ZF+DC that a countably generated σ -algebra can have a cardinality strictly larger than that of the continuum? Is it a theorem of ZF+DC that there exists an injective map from the set ω1 of well-orderings of N into R ? Thanks. 回答 略す
使用例 このような公理が無いとしても、各 n について普通の帰納法によって最初の n 項を有限列としてとることはできる。 従属選択公理が主張しているのは、その極限であるような可算無限列が取れるということである。 公理 DC は AC の断片であって、超限帰納法の各ステップで選択をする必要があって、それまでの選択に独立した選択ができない場合に、可算長の列を構成するのに必要である。
en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem History It turned out, though, that in first-order logic the well-ordering theorem is equivalent to the axiom of choice, in the sense that the Zermelo–Fraenkel axioms with the axiom of choice included are sufficient to prove the well-ordering theorem, and conversely, the Zermelo–Fraenkel axioms without the axiom of choice but with the well-ordering theorem included are sufficient to prove the axiom of choice. (The same applies to Zorn's lemma.) In second-order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering theorem.[7]
なお、"可算選択公理無し"の話は、下記のen.wikipedia Cauchy sequence で ”Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice” とあるので、ここまでは可です
ja.wikipedia.org/wiki/%E3%82%B3%E3%83%BC%E3%82%B7%E3%83%BC%E5%88%97 コーシー列 点列 (xn) が 略 数列の場合と同じく点列がコーシー的であるなどという これは、座標の各成分が全てコーシー数列を成すことと等価である また、やはり数列の場合と同様に、Rk における点列 (xn) がコーシー性を持つならば、十分大きな番号 n に対応する点 xn は例外なく全て、ある非常に小さな直径を持つ k 次元球体に含まれる 複素数全体の集合 C を座標平面 R2 と同一視してガウス平面と考えれば、複素数列は平面上の点の列であり、複素空間 Ck 内のコーシー列も同様に考えることができる
en.wikipedia.org/wiki/Cauchy_sequence Cauchy sequence Modulus of Cauchy convergence Any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The existence of a modulus for a Cauchy sequence follows from the well-ordering property of the natural numbers The existence of a modulus also follows from the principle of countable choice. Moduli of Cauchy convergence are used by constructive mathematicians who do not wish to use any form of choice. Using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Regular Cauchy sequences were used by Bishop (2012) and by Bridges (1997) in constructive mathematics textbooks.
In a metric space Since the definition of a Cauchy sequence only involves metric concepts, it is straightforward to generalize it to any metric space X.
Completeness A metric space (X, d) in which every Cauchy sequence converges to an element of X is called complete.
下記ですね ”When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω.”
(参考) en.wikipedia.org/wiki/Axiom_of_countable_choice Axiom of countable choice
Applications For instance, in order to prove that every accumulation point x of a set S⊆R is the limit of some sequence of elements of S∖{x}, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω.
下記で ”assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers.” google訳 ”可算選択公理を前提とすると、集合の濃度（集合の要素の数）が自然数の濃度より大きくない場合、その集合は可算です。有限でない可算集合は可算無限であると言われます。”
これ 百回音読してね ；ｐ）
(参考) https://en.wikipedia.org/wiki/Countable_set Countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.[a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
In more technical terms, assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite.
>>129より再録 ”assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers.”
なので、”assuming the axiom of countable choice”を採用します つまり、可算選択公理より、可算整列定理が従います
(参考) en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem Proof of axiom of choice The axiom of choice can be proven from the well-ordering theorem as follows. To make a choice function for a collection of non-empty sets, E, take the union of the sets in E and call it X. There exists a well-ordering of X; let R be such an ordering. The function that to each set S of E associates the smallest element of S, as ordered by (the restriction to S of) R, is a choice function for the collection E.■ An essential point of this proof is that it involves only a single arbitrary choice, that of R; applying the well-ordering theorem to each member S of E separately would not work, since the theorem only asserts the existence of a well-ordering, and choosing for each S a well-ordering would require just as many choices as simply choosing an element from each S.
イタリア版 （google英訳） it.wikipedia.org/wiki/Teorema_del_buon_ordinamento Well-ordering theorem Dependence of the axiom of choice We show that if every set is well-orderable, the axiom of choice holds. Given a family F, we would like to find a function f:F→∪X∈F X such that ∀X∈F,f(X)∈X. But on ∪X∈F X we can establish a well order < . Then, by the definition of well order, given a set X∈F, which will be a subset of ∪X∈F X we can find a minimal element.
The functionf(X)=min{y∈(X,<)} is a good choice function, since it is defined for each X and f(X)∈X. (引用終り)
証明のポイントは、 ”For every ordinal α, define an element aα that is in A by setting aα=f(A∖{aξ∣ξ<α}) ” の部分です。aα=f(A∖{aξ∣ξ<α})の部分が、選択公理における選択関数を成す A∖{aξ∣ξ<α}が集合族で、選択関数の定義域ですね
(参考) en.wikipedia.org/wiki/Well-ordering_theorem Well-ordering theorem 整列可能定理 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}.■
(参考) alg-d.com/math/ac/countable_union.html 可算和定理 壱大整域 命題「可算個の可算集合の和集合は可算集合」を可算和定理という．可算和定理は選択公理が無ければ証明できない． 証明 M を ZFC+GCH の可算推移的モデルとする．以下を満たす関数 p 全体がなす集合を P とする． 以下略
(参考) ja.wikipedia.org/wiki/%E6%95%B4%E5%88%97%E9%9B%86%E5%90%88 整列集合 導入 自然数全体の成す集合 N が通常の大小関係 "<" に関して整列集合となるという事実は、一般に整列原理と呼ばれる。 （選択公理に同値な）整列可能定理は、任意の集合が整列順序付け可能であることを主張するものである。整列可能定理はまたツォルンの補題とも同値である
en.wikipedia.org/wiki/Well-order Well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total ordering on S with the property that every non-empty subset of S has a least element in this ordering. The observation that the natural numbers are well ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers).
en.wikipedia.org/wiki/Well-ordering_principle Well-ordering principle In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element.[1] Properties Depending on the framework in which the natural numbers are introduced, this (second-order) property of the set of natural numbers is either an axiom or a provable theorem. For example:
(参考) https://en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice Results requiring AC (or weaker forms) but weaker than it ・Set theory ・The union of any countable family of countable sets is countable (this requires countable choice but not the full axiom of choice). （google訳） AC（またはより弱い形式）を必要とするが、それよりも弱い結果 ・集合論 ・可算集合の任意の可算族の和集合は可算です (これには可算な選択が必要ですが、選択公理の完全版は必要ありません)。
