これに戻る １）まず、”ZF上で実数は定義不可能”か？ について ”実数”の意味を明確にしておく必要があるが、それを カントールの集合論における”実数”と規定する つまり、下記に出てくる 実数の連続性（実数の完備性 (completeness of the real numbers) とも）を、備えたものとする ２）そうすると、下記 いろいろ辿ると ”Choice principles in elementary topology and analysis Horst Herrlich”(1997) にたどり着いて、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, " "5. R is a Lindel¨ of space, " "9. the Axiom of Choice for countable collections of subsets of R." "Equivalent are: " だと。つまり、"the Axiom of Choice for countable collections of subsets of R."でも " 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, " "R is a Lindel¨ of space, " までしか言えない、これが限界 （”Lindel¨ of”リンデレーエフは、下記ご参照） ３）ということは、"the Axiom of Choice for countable collections of subsets of R."を否定してしまうと ”実数”の連続性（実数の完備性）どころか、Lindelöfさえいえない。”in R, a point x”と”iff there exists a sequence in A＼{x} that converges to x, ” との関係も言えない
結論：(ZFCではなく)ZF上で実数の定義では、カントールの集合論の”実数”には、到達しない 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, これが限界です 従属選択公理で、実数の連続性（実数の完備性）が言えるか（フルパワー選択公理でなく）
(参考) en.wikipedia.org/wiki/Axiom_of_choice Axiom of choice Independence Many theorems provable using choice are of an elegant general character: the cardinalities of any two sets are comparable
Statements implying the negation of AC There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. As any model of ZF¬C is also a model of ZF, it is the case that for each of the following statements, there exists a model of ZF in which that statement is true. ・There is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i.e., for any sequence {xn} converging to a, limn f(xn)=f(a). ・The real numbers are a countable union of countable sets.[39] This does not imply that the real numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the Axiom of countable choice. つづく
en.wikipedia.org/wiki/Axiom_of_countable_choice Axiom of countable choice（ACω） 可算選択公理 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. Relation to other axioms Weaker systems Paul Cohen showed that ACω is not provable in Zermelo–Fraenkel set theory (ZF) without the axiom of choice.[6] Equivalent forms
fr.wikipedia.org/wiki/Axiome_du_choix_d%C3%A9nombrable Axiome du choix dénombrable 仏語 可算選択の公理
Par exemple, afin de prouver que tout point d'accumulation x d'un ensemble S⊆R est la limite d'une suite d'éléments de S\{x}, on a besoin (d'une forme faible) de l'axiome du choix dénombrable. Lorsqu'il est formulé pour les points d'accumulation d'espaces métriques arbitraires, l'énoncé devient équivalent à ACω3. （google訳） たとえば、集合S ⊆ Rの累積点xがS \{ x }の要素シーケンスの極限であることを証明するには、可算選択公理の (弱い形式) が必要です。任意の計量空間の累積点について定式化すると、このステートメントは AC ω 3と等価になります。
There exist models of ZF that violate the above conditions ([17], [18]). Observe the fine distinction between
89 名前：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 i. 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.
Notes et références 3．Pour d'autres énoncés équivalents à ACω, voir (en) Horst Herrlich, « Choice principles in elementary topology and analysis », Comment. Math. Univ. Carolinae, vol. 38, no 3,‎ 1997, p. 545-552 (lire en ligne [archive]) et (en) Paul Howard et Jean E. Rubin, Consequences of the Axiom of Choice, Providence, R.I., AMS, 1998.
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. Proposition 1.2 ([15]). Equivalent are: 1. in R, every bounded infinite set contains a convergent injective sequence, 2. every infinite subset of R is Dedekind-infinite. There exist models of ZF that violate the above conditions ([18]). Obviously, the conditions of Theorem 1.1 imply the conditions of Proposition 1.2. Is the converse true? Observe that the following slight modifications of condition 1 in Proposition 1.2 hold in ZF: (a) in R, every bounded countable set contains a convergent injective sequence, (b) in R, for every bounded infinite set there exists an accumulation point.
＜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
