”The axiom of dependent choice and no infinite descending sequence of sets implies regularity”（下記） な せめて選択公理の弱いバージョンが必要だってことな
https://en.wikipedia.org/wiki/Axiom_of_regularity Axiom of regularity
Contents 1 Elementary implications of regularity 1.2 No infinite descending sequence of sets exists 2 The axiom of dependent choice and no infinite descending sequence of sets implies regularity
The axiom of dependent choice and no infinite descending sequence of sets implies regularity
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by aRb:⇔ b∈ S∩a, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.
375 名前：2:46:58.61 ID:22hGBGwX.net mailto: >>316 >つまり、”called non-standard natural numbers”を含む列は、基礎の公理には反しないと説明されているよ 大間違い。 The hereditarily finite sets, Vω, satisfy the axiom of regularity (and all other axioms of ZFC except the axiom of infinity). So if one forms a non-trivial ultrapower of Vω, then it will also satisfy the axiom of regularity. The resulting model will contain elements, called non-standard natural numbers 遺伝的有限集合Vωは基礎の公理（及び無限公理を除く他のすべてのZFCの公理）を満たす。よって誰かがVωの非自明な超冪を形成したら、それも基礎の公理を満たす。結果モデルは超準数と呼ばれる要素を含むだろう。
ほいよ、下記 ”New Foundations has a universal set, so it is a non-well-founded set theory.[2] That is to say, it is an axiomatic set theory that allows infinite descending chains of membership such as … xn ∈ xn-1 ∈ … ∈ x2 ∈ x1.”
https://en.wikipedia.org/wiki/New_Foundations New Foundations New Foundations has a universal set, so it is a non-well-founded set theory.[2] That is to say, it is an axiomatic set theory that allows infinite descending chains of membership such as … xn ∈ xn-1 ∈ … ∈ x2 ∈ x1. It avoids Russell's paradox by permitting only stratifiable formulas to be defined using the axiom schema of comprehension. For instance x ∈ y is a stratifiable formula, but x ∈ x is not.
Contents 8 Models of NFU 8.1 Self-sufficiency of mathematical foundations in NFU 9 Strong axioms of infinity
https://ja.wikipedia.org/wiki/%E6%96%B0%E5%9F%BA%E7%A4%8E%E9%9B%86%E5%90%88%E8%AB%96 新基礎集合論 数理論理学において新基礎集合論 (しんきそしゅうごうろん、英: New Foundations) またはNF集合論とは、プリンキピア・マテマティカの型理論を単純化したものとしてウィラード・ヴァン・オーマン・クワイン[1]によって考案された、公理的集合論の一種である。この名称は、クワインが1937年における記事『数理論理学の新基礎』において初めて提唱したことに由来する。現在広く受け入れられているのはクワインが提唱したもともとの体系NFを少し修正したNFUと呼ばれる体系[2]である。
追加 （参考）”Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.[6] ” https://en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory Urelement Contents 1 Theory 2 Urelements in set theory Urelements in set theory The Zermelo set theory of 1908 included urelements, and hence is a version we now call ZFA or ZFCA (i.e. ZFA with axiom of choice).[1] It was soon realized that in the context of this and closely related axiomatic set theories, the urelements were not needed because they can easily be modeled in a set theory without urelements.[2] Thus, standard expositions of the canonical axiomatic set theories ZF and ZFC do not mention urelements. (For an exception, see Supp
382 名前：es.[3]) Axiomatizations of set theory that do invoke urelements include Kripke?Platek set theory with urelements, and the variant of Von Neumann?Bernays?Godel set theory described by Mendelson.[4] In type theory, an object of type 0 can be called an urelement; hence the name "atom." Adding urelements to the system New Foundations (NF) to produce NFU has surprising consequences. In particular, Jensen proved[5] the consistency of NFU relative to Peano arithmetic; meanwhile, the consistency of NF relative to anything remains an open problem, pending verification of Holmes's proof of its consistency relative to ZF. Moreover, NFU remains relatively consistent when augmented with an axiom of infinity and the axiom of choice. Holmes further argues that set theory is more natural with than without urelements, since we may take as urelements the objects of any theory or of the physical universe.[6]
https://en.wikipedia.org/wiki/Axiom_of_regularity Axiom of regularity
Regularity in the presence of urelements Urelements are objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other set theories such as ZFA, there are. In these theories, the axiom of regularity must be modified. The statement "{\displaystyle x\not =\emptyset }{\displaystyle x\not =\emptyset }" needs to be replaced with a statement that {\displaystyle x}x is not empty and is not an urelement. One suitable replacement is {\displaystyle (\exists y)[y\in x]}{\displaystyle (\exists y)[y\in x]}, which states that x is inhabited. (引用終り)
(参考) https://en.wikipedia.org/wiki/Infinity Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number.[1] It is often denoted by the infinity symbol shown here.
https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Infinite.svg/300px-Infinite.svg.png The infinity symbol
Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number.[1] It is often denoted by the infinity symbol shown here.
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol[2] and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hopital and Bernoulli)[3] regarded as infinitely small quantities, but infinity continued to be associated with endless processes.[4] As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.[2] At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.[2][5] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers.[6]
In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
In philosophy and theology, infinity is explored in articles under headings such as the Absolute, God, and Zeno's paradoxes.
In Greek philosophy, for example in Anaximander, 'the Boundless' is the origin of all that is. He took the beginning or first principle to be an endless, unlimited primordial mass (?πειρον, apeiron). The Jain metaphysics and mathematics were the first to define and delineate different "types" of infinities. The work of the mathematician Georg Cantor first placed infinity into a coherent mathematical framework. Keenly aware of his departure from traditional wisdom, Cantor also presented a comprehensive historical and philosophical discussion of infinity.[1]
https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Droste_1260359-nevit%2C_corrected.jpg/300px-Droste_1260359-nevit%2C_corrected.jpg Philosophers have speculated about the nature of infinity. Pictured is a simulation of the Droste effect.
max は存在するとは限りませんが，\supsup は(空でない場合は)常に存在するので，統一的に議論することができます。 \supsup の存在証明は解析学の教科書を参照して下さい（例えば高木貞治の解析概論）。
supが存在する条件として「 AA が空でない」が必要でした。ご指摘いただいた読者の方，ありがとうございます！
（本格的には下記などご参照） https://en.wikipedia.org/wiki/Infimum_and_supremum Infimum and supremum
Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves. (引用終り) 以上 []
意味わからん 後の引用 https://en.wikipedia.org/wiki/Infimum_and_supremum Infimum and supremum Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves. を見落とした？
１．無限下降列と無限上昇列との両方を禁止したら、それまずいよ（全てが有限列になる） ２．つまり、ZFCでやりたいのは、無限の高層ビルみたいなこと（即ち無限上昇列（それ当たり前))で、禁止したいのは「底なし沼」のようなどこまでも下降する列だよ（「底なし沼」は不要だと） ３．下記のOrdinal numberのポンチ絵 https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Ordinal_ww.svg/384px-Ordinal_ww.svg.png A graphical "matchstick" representation of the ordinal ω^2. Each stick corresponds to an ordinal of the form ω・m+n where m and n are natural numbers. を見てください。自然数の0,1,2・・が並んだ後に最小のthe first infinite ordinal, ωが来て、ω+1, ω+2, ω+3・・とつづく 0,1,2・・,ω,ω+1, ω+2, ω+3・・ （無限上昇列存在）と説明されている ４．ノイマンの基数割り当てでは、自然数Ｎ、つまり加算無限基数ｱﾚﾌ0 が、ω(=ω0)でもある（下記） ５．そして、ノイマンの基数割り当てでは、 ”0∈1∈2・・∈ω∈ω+1∈ ω+2∈ ω+3・・”でもある（分からない人は、下記”von Neumann cardinal assignment”を嫁め） ６．無限上昇列があれば、逆に辿れば、当然無限下降列になる。当たり前。それ禁止したらまずい その説明が、既に述べた >>316 https://en.wikipedia.org/wiki/Axiom_of_regularity Axiom of regularity - No infinite descending sequence of sets exists にあるよ ７．”「"最後の項がある"無限上昇列は存在しない」と言っている”は、単に、おサルがキーキー言っているだけの独自説にすぎない！（＾＾；
（参考） https://en.wikipedia.org/wiki/Ordinal_number Ordinal number
https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Ordinal_ww.svg/384px-Ordinal_ww.svg.png A graphical "matchstick" representation of the ordinal ω^2. Each stick corresponds to an ordinal of the form ω・m+n where m and n are natural numbers.
Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω・2 (which is ω+ω), ω・2+1, ω・2+2, and so on, then ω・3, and then later on ω・4. Now the set of ordinals formed in this way (the ω・m+n, where m and n are natural numbers) must itself have an ordinal associated with it: and that is ω^2. Further on, there will be ω^3, then ω^4, and so on, and ω^ω, then ω^ω^ω, then later ω^ω^ω^ω, and even later ε0 (epsilon nought) (to give a few examples of relatively small?countable?ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1 or Ω .[4][5][6]
（再録）ノイマンの基数割り当てでは、自然数Ｎ、つまり加算無限基数 ｱﾚﾌ0 が、ω0（=ω）でもある（下記） （”A graphical "matchstick" representation of the ordinal ω^2. Each stick corresponds to an ordinal of the form ω・m+n where m and n are natural numbers.”）
https://en.wikipedia.org/wiki/Von_Neumann_cardinal_assignment von Neumann cardinal assignment
Initial ordinal of a cardinal The α-th infinite initial ordinal is written Ω_α. Its cardinality is written ｱﾚﾌ_α (the α-th aleph number).
空集合を要素とし、任意の要素 x に対して x ∪ {x} を要素に持つ集合が存在する： ∃ A(Φ ∈ A∧∀ x∈ A(x∪{x}∈ A)) 解釈と帰結 上記定義では「無限」という言葉は用いられていないが、この公理によって（少なくとも1つの）無限集合の存在が保証されることになる。 まず定義中の集合 A は以下の性質を満たすことを確認できる。 Φ ∈ A（空集合 Φ は A の要素である） Φ ∪{Φ}={Φ}∈ A （「空集合 Φ を要素にもつ集合」は A の要素である） {Φ}∪{Φ ∪{Φ}}={Φ ,{Φ}}∈ A} （「空集合」と「空集合を要素にもつ集合」の2つを要素にもつ集合は A の要素である） （以下同様に繰り返す） 各手続きで得られた集合を要素とする集合を B:={Φ ,{Φ},{Φ ,{Φ}},・・・} とおくと、 B は A の部分集合である。 この手続きは何回でも繰り返すことができるが、もし有限回で終えた場合、 B は有限集合であり、 A≠ Bである。なぜならば定義により B∪{B}∈ A であるが、 B∪{B} not∈ B となるからである。一方 A が有限集合であれば、この手続きを繰り返すことで B が A よりも多くの要素をもつことができてしまう。 従って A は有限集合ではない（すなわち無限集合である）ため、無限公理を採用すれば直ちに無限集合の存在を認めることになる。
>>380 > A graphical "matchstick" representation of the ordinal ω^2. Each stick corresponds to an ordinal of the form ω・m+n where m and n are natural numbers. > を見てください。自然数の0,1,2・・が並んだ後に最小のthe first infinite ordinal, ωが来て、ω+1, ω+2, ω+3・・とつづく それ、列じゃないからｗ 列だったらωの前者が居ないとダメｗ ホントバカだね
定義 集合あるいはクラス X 上の二項関係 R が整礎であるとは、X の空でない任意の部分集合 S が R に関する極小元を持つことをいう[1]。（関係 R がさらに集合的であることを仮定する著者もいる[2]。X が集合であればこれは自動的に成り立つ。）つまり、S の元 m であって、S の任意の元 s に対して対 (s, m) は R に属さないようなものが存在する。式で書けば
∀ S⊆ X(S≠ Φ → exists min S;;∀ sin S;,(s,m)notin R).} ∀ S⊆ X;,(S≠ Φ → exists min S;;∀ sin S;,(s,m)notin R).} X が集合であるとき、従属選択公理（英語版）（これは選択公理よりも真に弱く可算選択公理よりも真に強い）を仮定すれば、同値な定義として、関係が整礎であることを可算無限降下列が存在しないこととして定められる[3]。
集合 x が整礎的集合 (well-founded set) であることは、∈ が x の推移閉包上で整礎関係となることと同値である。ZF における公理のひとつである正則性の公理は、全ての集合が整礎であることを要請するものである。
関係 R が X 上で逆整礎 (converse well-founded) または上方整礎 (upwards well-founded) であるとは、R の逆関係 R^-1 が X 上の整礎関係であるときにいう。
帰納法と再帰 整礎関係が興味深い重要な理由は、それによって超限帰納法の一種が考えられることにある。すなわち (X, R) が整礎関係で P(x) が X の元に関する何らかの性質であるときに、 P(x) が X の「すべての」元に対して満たされることを示すには、以下を示せば十分である。
x を X の元とするとき、y R x なる全ての y に対して P(y) が真であるならば P(x) は必ず真である。つまり、 略 が成り立つ。 このような整礎帰納法 (well-founded induction) は、エミー・ネーターにちなんでネーター帰納法 (Noetherian induction) とも呼ばれることがある[4]。
帰納法と同様に、整礎関係は超限再帰による対象の構成も保証する。(X, R) が集合的整礎関係で F が X の元 x と X の始切片 {y | y R x} 上の函数 g の組に対して対象 F(x, g) を割り当てる函数とすると、函数 G が一意的に存在して、任意の x ∈ X に対して G(x)=F(x,G｜y|yRx) が満たされる。つまり、X 上の函数 G を構成しようとするとき、G(x) を y R x なる y に対する値 G(y) を利用して定義することができる。
例として、整礎関係 (N, S) を考える。ここで N は自然数全体のなす集合で、S は後者函数 x → x + 1 のグラフとする。S 上の帰納法は通常の数学的帰納法であり、S 上の再帰は原始再帰を与える。順序関係 (N, <) からは完全帰納法 (complete induction) と累積帰納法 (course-of-values recursion) が得られる。 (N, <) が整礎関係であるという言明は整列原理としても知られる。
その他の性質 (X, <) が整礎関係で x が X の元ならば、x から始まる降鎖列は必ず長さ有限だが、これはこのような降鎖の長さが有界であるということを意味しない。 以下のような例を考えよう。X は正の整数全体の成す集合に、どの整数よりも大きな整数ではない新しい元 ω を付け加えた集合とする。 このとき X は整礎だが、ω から始まる長さ有限の降鎖列でいくらでも長いものが取れる。なんとなれば、任意の正整数 n に対して ω, n - 1, n - 2, ..., 2, 1 という鎖は長さ n を持つ。
モストウスキーの崩壊補題 (Mostowski collapse lemma) によれば、集合要素関係 (set membership) は普遍的な整礎関係である。 つまり、クラス X 上の集合的な整礎関係 R に対し、クラス C が存在して、(X, R) が (C, ∈) に同型となる。
反射関係の整礎性 関係 R が反射律を満たすとは、R の始域の任意の元 a に対して a R a が満たされることである。任意の定値列は（広義の）降鎖であるから、始域が空でない任意の反射関係は無限降鎖をもつ。例えば、自然数の全体に通常の大小関係による順序 ≦ を考えれば 1 ≧ 1 ≧ 1 ≧ ? は無限降鎖になる。反射関係 R を扱う際には、この手の自明な降下列を取り除くために、普通は（しばしば陰伏的に） a R′b ⇔ a R b かつ a ≠ b で定義される関係 R′ を代わりに利用する。先ほどの自然数の例で言えば、反射的順序関係 ≦ を考える代わりに、整礎関係となる < を用いるということである。
In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x. The axiom of regularity, which is one of the axioms of Zermelo?Fraenkel set theory, asserts that all sets are well-founded.
A relation R is converse well-founded, upwards well-founded or Noetherian on X, if the converse relation R^-1 is well-founded on X. In this case R is also said to satisfy the ascending chain condition. In the context of rewriting systems, a Noetherian relation is also called terminating.
There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction. When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details.
Other properties If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n - 1, n - 2, ..., 2, 1 has length n for any n.
The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X, R) is isomorphic to (C, ∈).
Application Every set model of ZF is set-like and extensional. If the model is well-founded, then by the Mostowski collapse lemma it is isomorphic to a transitive model of ZF and such a transitive model is unique.
Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model. There exists a model M (assuming the consistency of ZF) whose domain has a subset A with no R-minimal element, but this set A is not a "set in the model" (A is not in the domain of the model, even though all of its members are). More precisely, for no such set A there exists x in M such that A = R-1[x]. So M satisfies the axiom of regularity (it is "internally" well-founded) but it is not well-founded and the collapse lemma does not apply to it.
https://en.wikipedia.org/wiki/Mostowski_collapse_lemma Mostowski collapse lemma Application Every set model of ZF is set-like and extensional. If the model is well-founded, then by the Mostowski collapse lemma it is isomorphic to a transitive model of ZF and such a transitive model is unique.
Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model. There exists a model M (assuming the consistency of ZF) whose domain has a subset A with no R-minimal element, but this set A is not a "set in the model" (A is not in the domain of the model, even though all of its members are). More precisely, for no such set A there exists x in M such that A = R-1[x]. So M satisfies the axiom of regularity (it is "internally" well-founded) but it is not well-founded and the collapse lemma does not apply to it. (引用終り) 以上
https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior Limit inferior and limit superior
The case of sequences of real numbers In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [-∞,∞], which is a complete lattice.
Properties As mentioned earlier, it is convenient to extend {\displaystyle \mathbb {R} }\mathbb {R} to [-∞,∞]. (Note that when working just in {\displaystyle \mathbb {R} }\mathbb {R} , convergence to -∞ or ∞ would not be considered as convergence.) (引用終り) 以上
>>414 ＞このとき X は整礎だが、ω から始まる長さ有限の降鎖列でいくらでも長いものが取れる。
下記レーヴェンハイム-スコーレムの定理より 「定理の上方部分の証明は、いくらでも大きな有限のモデルを持つ理論は無限のモデルを持たねばならないことをも示す」です (”The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model”)
https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem Lowenheim-Skolem theorem Consequences The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem. (引用終り) 以上
一階述語論理と同様に議論領域（ドメイン）の考え方を使う。ドメインとは、量化可能な個々の元の集合である。一階述語論理では、そのドメインの個々の元が変項の値となり、量化される。例えば、一階の論理式 ∀x (x ≠ x + 1) では、変項 x は任意の個体を表す。二階述語論理は個体の集合を変項の値とし、量化することができる。例えば、二階の論理式 ∀S ∀x (x ∈ S ∨ x ? S) は、個体の全ての集合 S と全ての個体 x について、x が S に属するか、あるいは属さないかのどちらかであるということを主張している。最も一般化された二階述語論理は関数の量化をする変項も含んでいる（詳しくは後述）。
https://en.wikipedia.org/wiki/Indeterminate_form Indeterminate form There are seven indeterminate forms which are typically considered in the literature:[2] 0/0,∞/∞ ,0ｘ∞ ,∞-∞ ,0^0,1^∞ ,and ∞^0. []
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： 1．|A| = |B| ⇔ A ～ B 。 2．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。 (引用終り) 以上
”レーヴェンハイム-スコーレム”補足参考（これ分かりやすいよ（＾＾ ） www.cs-study.com/koga/set/lowenheimSkolem.html 形式的論理体系の定義から レーベンハイム・スコーレムの定理までの大急ぎのまとめ (Rapid Summary from Syntax of Logic to Lowenheim-Skolem Theorem) by Akihiko Koga 27th Mar. 2020 (Update) 13th July 2018 (First)
関連追加メモ https://en.wikipedia.org/wiki/Alternative_set_theory Alternative set theory
In a general sense, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo?Fraenkel set theory. More specifically, Alternative Set Theory (or AST) may refer to a particular set theory developed in the 1970s and 1980s by Petr Vop?nka and his students.
Vop?nka's Alternative Set Theory Vop?nka's Alternative Set Theory builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo?Fraenkel (or ZF) set theory, in which the axiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor (ZF)
505 名前：finite sets and they are called infinite in AST.
Other alternative set theories Other alternative set theories include:[1] ・Von Neumann?Bernays?Godel set theory ・Morse?Kelley set theory ・Tarski?Grothendieck set theory ・Ackermann set theory ・Type theory ・New Foundations ・Positive set theory ・Internal set theory ・Naive set theory ・S (set theory) ・Kripke?Platek set theory ・Scott?Potter set theory ・Constructive set theory
See also ・Non-well-founded set theory ・List of first-order theories § Set theories []
「集合xに無限下降列が存在してもxは正則性公理に反しない」 を正当化しようとして初歩的誤りを重ね続ける瀬田くん。 なぜ間違いを認められない幼稚な性格なのか？ 私なんてあっさり認めたぞ？ｗ>>468 数学Dr. Prussだって認めたぞ？ｗ What we have then is this: For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right.
それはさておき、”数学Dr. Prussだって認めたぞ？ What we have then is this: For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right.” は違うよ
その話は、スレチだが下記な 箱入り無数目を語る部屋 https://rio2016.5ch.net/test/read.cgi/math/1609427846/52-56 https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice mathoverflow Probabilities in a riddle involving axiom of choice asked Dec 9 '13 at 16:16 Denis ＜回答12の質疑応答＞ What we have then is this: For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right. But now the question is whether we can translate this to a statement without the conditional "For each fixed opponent strategy". ? Alexander Pruss Dec 19 '13 at 15:05
まず >That's right. But now the question 典型的な「イエスバット法」（下記）でしょ 会話の基本テクニック ある程度相手の言い分を認めつつ、自分の主張を展開するのです ”But”以下に力点がありますよ
さらに、”if i is chosen uniformly independently”とあるところにご注目 これ、https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice Pruss氏の12回答本文の ”Let's go back to the riddle. Suppose u^→ is chosen randomly. The most natural option is that it is a nontrivial i.i.d. sequence (uk), independent of the random index i which is uniformly distributed over [100]={0,...,99}. In general, Mj will be nonmeasurable (one can prove this in at least some cases). We likewise have no reason to think that M is measurable. But without measurability, we can't make sense of talk of the probability that the guess will be correct.” を再度強調している部分
534 名前：ですよ ”uniformly”の部分は、明らかに ”independent of the random index i which is uniformly distributed over [100]={0,...,99}”のところを、短く言い直しています そして、本文にあるように、もし、” M is measurable”なら、問題の戦略は正しいが、 実際は”But without measurability, we can't make sense of talk of the probability that the guess will be correct.”ってことです 問題の”riddle”は、一見は、一様分布 {0,...,99}とか{0,...,n}に見えるけれども、実際はそうではない。だから”riddle”（ナゾナゾ）ってことです
（参考） https://en.wikipedia.org/wiki/Uniform_distribution Uniform distribution Uniform distribution may refer to: Continuous uniform distribution Discrete uniform distribution
https://en.wikipedia.org/wiki/Discrete_uniform_distribution In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen".
A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability.
https://upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Uniform_discrete_pmf_svg.svg/488px-Uniform_discrete_pmf_svg.svg.png discrete uniform Probability mass function
数学の位相空間論周辺分野において、位相空間 X の部分集合 A が X において稠密（ちゅうみつ、英: dense）であるとは、X の各点 x が、A の元であるか、さもなくば A の集積点であるときにいう[1]。イメージで言えば、X の各点が A の中か、さもなくば A の元の「どれほどでも近く」にあるということを表している。例えば、有理数は実数の稠密集合である。なぜなら任意の実数は、有理数であるか、さもなくばどれほどでも近い有理数をとることができるからである（ディオファントス近似も参照）。
目次 1 厳密な定義 2 距離空間における稠密性 3 例 4 性質 5 関連概念
関連概念 位相空間 X の部分集合 A の点 x が、X における A の極限点（limit point）であるとは、x の任意の近傍が x 以外に少なくとも一つ A の元を含むときにいう[3]。さもなくば、x を A の孤立点（isolated point）という[4]。
>>487 >さらに、”if i is chosen uniformly independently”とあるところにご注目 それは箱入り無数目でいうところの「さて， 1～100 のいずれかをランダムに選ぶ. 」のところですよｗ 文章の切り方が変ですよ？ independently を書くなら�
547 名前：ｻの後ろも書かないと意味が通じませんよ？ｗ independently of that strategy とは、出題実数列とは独立にという意味ですｗ iidとは何の関係もありませんｗ まあ蛇足なんですけどねｗ uniformly とは一様分布で、つまりランダムに、という意味で、必然出題実数列とは独立になりますからｗ
>これ、https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice >Pruss氏の12回答本文の >”Let's go back to the riddle. Suppose u^→ is chosen randomly. The most natural option is that it is a nontrivial i.i.d. sequence (uk), independent of the random index i which is uniformly distributed over [100]={0,...,99}. In general, Mj will be nonmeasurable (one can prove this in at least some cases). We likewise have no reason to think that M is measurable. But without measurability, we can't make sense of talk of the probability that the guess will be correct.” これはThe Riddle でも The Modification でもなく、箱の中身を確率変数とする戦略の話。 箱の中身を確率変数としていない時枝戦略とは何の関係もありませんｗ
>を再度強調している部分ですよ いいえ、無関係ですね。 independent of the random index i which is uniformly distributed over [100]={0,...,99} が読めないんですか？random index i とは「 独 立 」と言ってるのになんで再度強調になるんですか？ｗ 文盲ですか？ｗ
>”uniformly”の部分は、明らかに ”independent of the random index i which is uniformly distributed over [100]={0,...,99}”のところを、短く言い直しています unifoirmly とは一様分布でという意味ですよｗ 辞書くらい調べましょうｗ []
>>487 >そして、本文にあるように、もし、” M is measurable”なら、問題の戦略は正しいが、 >実際は”But without measurability, we can't make sense of talk of the probability that the guess will be correct.”ってことです 箱の中身を確率変数としていない時枝戦略とは何の関係も無いですねｗ 勝つ戦略の存在性は時枝戦略で尽くされてますｗ 箱の中身を確率変数とする戦略が勝つ戦略でないとしても（実際確率空間が非可測になるので勝つ戦略でない）、勝つ戦略の存在性には何ら影響ありませんｗ
>問題の”riddle”は、一見は、一様分布 {0,...,99}とか{0,...,n}に見えるけれども、実際はそうではない。 大間違いですねｗ The Riddle はそもそも確率を一切使いませんｗ 必然確率空間の可測性は無関係ですｗ The Modification 及び時枝戦略は確率を使いますが、箱の中身を確率変数とはしていないのでやはり確率空間の可測性は関係ありませんｗ 具体的に言うと確率変数は random index i which is uniformly distributed over [100]={0,...,99} であり、離散一様分布であり、その確率空間可測ですｗ
導入 整列集合 X の任意の元 s は、それが X の最大元でない限り、ただ一つの後者（successor; 後継、次の元、直後の元）を持つ。これはつまり、s よりも大きな X の元全体の成す部分集合における最小元として s の後者が決まるということである。また、整列集合 X の中で上に有界な任意の部分集合は（その上界全体の成す X の部分集合に最小元がとれるから）必ず上限を持つ。あるいは整列集合 X には、前者（predecessor; 直前の元）を持たない元が必ず存在する（それはもちろん、X 全体における最小元である）。 集合に整列順序が与えられれば、そこでは集合の全ての元に対する命題の超限帰納法を用いた証明を考えることができる。 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。
13.1 整列集合 順序集合 (X, ?) は, すべての空でない部分集合が最小元をもつとき, 整列集 合であるといい, そのような順序を整列順序という. 定義から整列集合は必ず全 順序集合であることに注意しよう. 実際, a, b ∈ X に対して集合 {a, b} は X の 空でない部分集合になるから, それは最小元をもつ. 最小元は a または b であ るが, それが a であれば a ? b となるし, それが b であれば b ? a となる. こ れは, 任意の a, b ∈ X が比較可能であることを意味し, X は全順序集合である ことがわかる. 定義から空でない整列集合 X それ自身は最小元 min X をもつ.
定 理 13.1 自然数 (N, ?) は整列集合である. 証 明 いつも通り, [n] = {1, 2, . . . , n} と書くことにする. A ⊂ N を空でない 部分集合とする. このとき, A = A ∩ N = A ∩∪n=1～∞[n] = ∪n=1～∞ A ∩ [n] と A≠ Φ から, A ∩ [n]≠ Φ を満たす n ∈ N が存在する. そこで, N の部分集合 A が A ∩ [n]≠ Φ を満たせば A は最小元をもつことを示せばよい. そのことを数学的帰納法で証明しよう. まず, n = 1 のときは A ∩ [1]≠ Φ か ら 1 ∈ A がわかる. 1 は自然数の中で最小であるから, 確かに A は最小元をも つ. 次に n ? 1 まで主張が正しいと仮定して, A ∩ [n + 1]≠ Φ とする. もし, A ∩ [n]≠ Φ であれば帰納法の仮定から A は最小元をもつ. A ∩ [n] = Φ であれ ば, A ∩ [n + 1]≠ Φ と合わせて n + 1 が A の最小元であることがわかる. (引用終り)
>>486 追加 >それはさておき、 >”数学Dr. Prussだって認めたぞ？ > What we have then is this: For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the >"independently" here > isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right.” >は違うよ
＜補足＞ 1．”if i is chosen uniformly independently of that strategy”の部分で 例えば、簡単にｎ列で、ｎ＜ｍの有限整数ｍで、1～ｍの札各1枚｛1,2,・・,ｍ｝で、 等確率（つまり一様分布（uniformly））で、各列1枚の数の札ni(i=1～n)を割り当てるとする ｎ枚の札で、max{n1,n2,・・ni・・,nn} になったら負けとする。つまり負けは、最大値の1列のみ。勝つ確率は、 普通に(n-1)/nとなる。 2．しかし、ｍ→∞とすると、一様分布（uniformly）でなくなる 下記の「非正則事前分布」で積分値が無限大に発散し、コルモゴロフの確率の公理に反します 3．さらに、時枝記事の決定番号は、”札各1枚”ではない！ つまり ”ni”を実現する代表列は複数あり、一様ではない分布を持ちます ですから、Pruss氏のいう一様分布（uniformly）を満たしません 4．結局、時枝記事の決定番号は、ｍ→∞となり積分が無限大に発散することと、 さらに ”ni”を実現する代表列は複数あり一様ではない分布を持つということと、 二重の意味で、”without measurability”です （この”without measurability”は、ビタリの意味の非可測とはちょっと違う。主に、積分が∞に発散することによる） 5．従って、Pruss氏の下記”But”以下の文に繋がります 6．結局、Pruss氏は、質問の”Probabilities in a riddle involving axiom of choice”の「 strategy 」の存在を、否定しています（下記 mathoverflowご参照） 以上
https://mathoverflow.net/questions/151286/probabilities-in-a-riddle-involving-axiom-of-choice mathoverflow Probabilities in a riddle involving axiom of choice asked Dec 9 '13 at 16:16 Denis ＜回答12の質疑応答＞ What we have then is this: For each fixed opponent strategy, if i is chosen uniformly independently of that strategy (where the "independently" here isn't in the probabilistic sense), we win with probability at least (n-1)/n. That's right. But now the question is whether we can translate this to a statement without the conditional "For each fixed opponent strategy". ? Alexander Pruss Dec 19 '13 at 15:05 (引用終り) 以上 []
つまり, DC とは, 極大要素を持たない二項関係は無限上昇鎖をもつ, という主張です. あきらかに, 選択公 理 AC は DC を導きます. 逆に DC から AC を導くことができ�
610 名前：ﾈいことは, 定理 1 によって明らかです*6. DC はルベーグ可測でない集合の存在を導くほどには強くないのです. そのいっぽうで, 測度の理論に必要となる, 可算個の集合からの同時選択 (可算選択の公理) は DC によっ て保証されます. また, 第 3 節で展開されるボレル集合のコードの理論には, 可算選択の公理だけでは不十分 で, 本当に DC が必要です. その理由は, DC が整礎的二項関係のとりあつかいを簡単にする点にあります.
定義 3. 二項関係 R が整礎的 (wellfounded) であるとは, R の定義域の空でない任意の部分集合 S が条件 ∃s ∈ S∀t ∈ S [t ≠ s =⇒ ht, si ∈/ R ] をみたす場合にいう. この条件にあらわれるような s は, S の R-極小 (minimal) な要素と呼ばれる. □
命題 3. (ZF) 集合 X 上の二項関係 R が整礎的であるための必要十分条件は, 順序数値関数 φ : X → ON が存在して ∀x, y ∈ X [hx, yi ∈ X =⇒ φ(x) < φ(y) ] をみたすことである. □ 命題 4. (ZF) 従属選択の公理 DC は次の命題と同値である: 集合 X 上の二項関係 R が整礎的でなければ, ωX の要素 f が存在して, すべての自然数 n について hf(n + 1), f(n)i ∈ R をみたす. つまり, R-無限下降 列が存在する. □ 注） *6 とはいえ定理 1 は到達不可能基数の存在に訴えるものですから, 厳密にいえばこの議論は AC の ZF + DC からの独立性の証明 にはなっていません. しかし, ZF が矛盾しなければ DC と ￢AC を付けくわえても矛盾しないというのは本当です. (引用終り) 以上 []
pervert (自動) いやらしい目つきで見る、変態的行為をする、いやらしい［エッチな］行為にふける (他動) 1.〔正しい道などを〕踏み外す、〔善などに〕背を向ける ・You must cease to pervert the right ways of the Lord. : 神の定めた道を踏み外すことをやめなければならない。 2.〔～を〕悪化させる、〔～を〕おとしめる ・Some people fear that these new technologies will pervert their values they have struggled so long to achieve. : これらの新しい技術が長い間かけて勝ち取ってきた価値観をおとしめることになると危ぶむ人もいる。 3.〔～を〕悪用する、〔～を〕不適切に用いる ・The military forces might pervert their power to injury of their fellow citizens. : 軍隊が自らの力を悪用して同胞を傷つけるようなことにならないとも限らない。 4.〔～を〕曲解する、〔～を〕誤解する ・Some think terrorists pervert the meaning of Islam. : テロリストはイスラム教の意味を曲解していると考える人もいる。 (名) 1.性的倒錯者、変質者◆【略】perv 2.背教者
（参考：Zermelo’s Axiomatization） https://plato.stanford.edu/entries/zermelo-set-theory/ Stanford Encyclopedia of Philosophy Zermelo’s Axiomatization of Set Theory First published Tue Jul 2, 2013 1. The Axioms The introduction to Zermelo's paper makes it clear that set theory is regarded as a fundamental theory:
VII.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 Φ, {Φ}, {{Φ}}, ….) With the inclusion of this last, Zermelo explicitly rejects any attempt to prove the existence of an infinite collection from other principles, as we find in Dedekind (1888: §66), or in Frege via the establishment of what is known as ‘Hume's Principle’. The four central axioms of Zermelo's system are the Axioms of Infinity and Power Set, which together show the existence of uncountable sets, the Axiom of Choice, to which we will devote some space below, and the Axiom of Separation.
There were attempts at the statement of axioms before Zermelo, both publicly and in private correspondence.[6] In particular, Cantor, in correspondence with Hilbert and Dedekind in the late 1890s, had endeavoured to describe some principles of set existence[7] which he thought were legitimate, and would not give rise to the construction of what he called ‘inconsistent totalities’, totalities which engender contradictions. (The best known of these totalities were the totality of all ordinals and the totality of all cardinals.) These principles included those of set union and a form of the replacement axiom, as well as principles which seem to guarantee that every cardinal number is an aleph, which we call for short the ‘Aleph Hypothesis (AH)’.
Despite this, there are reasons for calling Zermelo's system the first rea
640 名前：l axiomatisation of set theory. It is clear above all that Zermelo's intention was to reveal the fundamental nature of the theory of sets and to preserve its achievements, while at the same time providing a general replacement for the CP.
>>578 つづき 2. The Background to Zermelo's Axiomatisation 2.1 Hilbert's Axiomatic Method
2.1.2 Proof analysis and Zermelo's Well-Ordering Theorem [WOT]
2.2 The Well-Ordering Problem and the Well-Ordering Theorem 2.2.1 The importance of the problem before Zermelo
2.2.5 The Axioms of the 1908 WOT Paper He also adds the Axiom of Infinity, to guarantee that there are infinite sets, and the Axiom of Extensionality, which codifies the assumption that sets are really determined by their members, and not by the accidental way in which these members are selected. In addition, as we have noted, he now calls the Axiom of Choice by this name.
3.2.3 Cardinality It was pointed out by both Fraenkel and Skolem in the early 1920s that Zermelo's theory cannot provide an adequate account of cardinality. The axiom of infinity and the power set axiom together allow the creation of sets of cardinality ≧ ｱﾚﾌn for each natural number n, but this (in the absence of a result showing that 2^ｱﾚﾌ0 > ｱﾚﾌn for every natural number n) is not enough to guarantee a set whose power is ≧ ｱﾚﾌω, and a set of power ｱﾚﾌω is a natural next step (in the Cantorian theory) after those of power ｱﾚﾌn. Fraenkel proposed a remedy to this (as did Skolem independently) by proposing what was called the Ersetzungsaxiom, the Axiom of Replacement (see Fraenkel 1922: 231 and Skolem 1923: 225?226). This says, roughly, that the ‘functional image’ of a set must itself be a set, thus if a is a set, then {F(x) : x ∈ a} must also be a set, where ‘F’ represents a functional correspondence. Such an axiom is certainly sufficient; assume that a0 is the set of natural numbers {0, 1, 2, …}, and now assume that to each number n is associated an an with power ｱﾚﾌn. Then according to the replacement axiom, a = {a0, a1, a2, …} must be a set, too. This set is countable, of course, but (assuming that the an are all disjoint) the union set of a must have cardinality at least ｱﾚﾌω. つづく
3.2.4 Ordinals Although Kuratowski's work solved many of the representational problems for Zermelo's theory, and the Replacement Axiom shows how the most obvious cardinality gap can be closed, there still remained the issue (Kuratowski's view to one side) of representing accurately the full extent of the theory which Cantor had developed, with the transfinite numbers as fully fledged objects which ‘mirror’ the size/ordering of sets. Once the ordinal number-classes are present, the representation of the alephs is not a severe problem, which means that the representation of transfinite numbers amounts to assuring the existence of sufficiently many transfinite ordinal numbers. Indeed, as was stated above, the hypothesis that the scale of aleph numbers is sufficient amounts to the claim that any set can be ‘counted’ by some ordinal. There are then two interrelated problems for the ‘pure’ theory of sets: one is to show how to define ordinals as sets in such a way that the natural numbers generalise; the other problem is to make sure that there are enough ordinals to ‘count’ all the sets.
The problem w
643 名前：as fully solved by von Neumann in his work on axiomatic set theory from the early 1920s. Cantor's fundamental theorems about ordinal numbers, showing that the ordinals are the representatives of well-ordered sets, are the theorem that every well-ordered set is order-isomorphic to an initial segment of the ordinals, and that every ordinal is itself the order-type of the set of ordinals which precede it. These results prove crucial in the von Neumann treatment. Von Neumann's basic idea was explained by him as follows:
What we really wish to do is to take as the basis of our considerations the proposition: ‘Every ordinal is the type of the set of all ordinals that precede it’. But in order to avoid the vague notion ‘type’, we express it in the form: ‘Every ordinal is the set of the ordinals that precede it’. (von Neumann 1923, p. 347 of the English translation)
According to von Neumann's idea, 1 is just {0}, 2 is just {0, 1}, 3 is just {0, 1, 2} and so on. On this conception, the first transfinite ordinal ω is just {0, 1, 2, 3, …, n, …}, and generally it's clear that the immediate successor of any ordinal α is just α ∪ {α}. If we identify 0 with Φ, as Zermelo did, then we have available a reduction of the general notion of ordinal to pure set theory, where the canonical well-ordering on the von Neumann ordinals is just the subset relation, i.e., α < β just in case α ⊂ β, which von Neumann later shows is itself equivalent to saying α ∈ β. (See von Neumann 1928, p. 328 of the reprinting.) So again, inclusion orderings are fundamental.
Von Neumann gives a general definition of his ordinals, namely that a set α is an ordinal number if and only if it is a set ordered by inclusion, the inclusion ordering is a well-ordering, and each element ξ in α equals the set of elements in the initial segment of the ordering determined by ξ. This connects directly with Kuratowski's work in the following way. Suppose M is a well-ordered set which is then mirrored by an inclusion chain M in the power set of M. Then the first few elements of the inclusion chain will be the sets Φ, {a}, {a, b}, {a, b, c}, …, where a, b, c, … are the first, second, third …elements in the well-ordering of M. The von Neumann ordinal corresponding to M will also be an inclusion ordering whose first elements will be
Φ, {Φ}, {Φ, {Φ}}, {Φ, {Φ}, {Φ, {Φ}}}, … (in other words, 0, 1, 2, 3…), and we have 0 ⊂ 1 ⊂ 2 ⊂ 3 ⊂… in mirror image of Φ ⊂ {a} ⊂ {a, b} ⊂ {a, b, c} ⊂ …
These von Neumann ordinals had, in effect, been developed before von Neumann's work. The fullest published theory, and closest to the modern account, is to be found in Mirimanoff's work published in 1917 and 1921 (see Mirimanoff 1917a,b, 1921), though he doesn't take the final step of identifying the sets he characterises with the ordinals (for an account of Mirimanoff's work, see Hallett 1984: 273?275). It is also clear that Russell, Grelling and Hessenberg were close to von Neumann's general set-theoretic definition of ordinals. But crucially Zermelo himself developed the von Neumann conception of ordinals in the years 1913?1916, (for a full account, see Hallett 1984: 277?280 and Ebbinghaus 2007: 133?134). Zermelo's idea was evidently well-known to the Gottingen mathematicians, and there is an account of it in Hilbert's lectures ‘Probleme der mathematischen Logik’ from 1920, pp. 12?15.[37]
Despite all these anticipations, it is still right to ascribe the theory to von Neumann. For it was von Neumann who revealed the extent to which a full theory of the ordinals depends
646 名前： on the Axiom of Replacement. As he wrote later:
A treatment of ordinal number closely related to mine was known to Zermelo in 1916, as I learned subsequently from a personal communication. Nevertheless, the fundamental theorem, according to which to each well-ordered set there is a similar ordinal, could not be rigorously proved because the replacement axiom was unknown. (von Neumann 1928: 374, n. 2)
The theorem von Neumann states is the central result of Cantor's mentioned here in the second paragraph of this section. As von Neumann goes on to point out here (also p. 374), it is the possibility of definition by transfinite induction which is key, and a rigorous treatment of this requires being able to prove at each stage in a transfinite inductive process that the collection of functional correlates to a set is itself a set which can thus act as a new argument at the next stage. It is just this which the replacement axiom guarantees. Once justified, definition by transfinite induction can be used as the basis for completely general definitions of the arithmetic operations on ordinal numbers, for the definition of the aleph numbers, and so on. It also allows a fairly direct transformation of Zermelo's first (1904) proof of the WOT into a proof that every set can be represented by (is equipollent with) an ordinal number, which shows that in the Zermelo system with the Axiom of Replacement added there are enough ordinal numbers.[38]
It is thus remarkable that von Neumann's work, designed to show how the transfinite ordinals can be incorporated directly into a pure theory of sets, builds on and coalesces with both Kuratowski's work, designed to show the dispensability of the theory of transfinite ordinals, and also the axiomatic extension of Zermelo's theory suggested by Fraenkel and Skolem.
https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem Lowenheim-Skolem theorem In mathematical logic, the Lowenheim-Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Lowenheim and Thoralf Skolem.
The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number ｋ it has a model of size ｋ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are un
717 名前：able to control the cardinality of their infinite models. In general, the Lowenheim-Skolem theorem does not hold in stronger logics such as second-order logic. Consequences The statement given in the introduction follows immediately by taking M to be an infinite model of the theory. The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem. Proof sketch Upward part First, one extends the signature by adding a new constant symbol for every element of M. The complete theory of M for the extended signature σ' is called the elementary diagram of M. In the next step one adds ｋ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for any two distinct new constant symbols c and c'. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least ｋ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly ｋ. It contains an isomorphic copy of M as an elementary substructure.[3][4]:100-102 In other logics Main article: Lowenheim number Although the (classical) Lowenheim-Skolem theorem is tied very closely to first-order logic, variants hold for other logics. For example, every consistent theory in second-order logic has a model smaller than the first supercompact cardinal (assuming one exists).
The minimum size at which a (downward) Lowenheim-Skolem-type theorem applies in a logic is known as the Lowenheim number, and can be used to characterize that logic's strength. Moreover, if we go beyond first-order logic, we must give up one of three things: countable compactness, the Downward Lowenheim-Skolem Theorem, or the properties of an abstract logic.[5]:134
https://en.wikipedia.org/wiki/L%C3%B6wenheim_number Lowenheim number In mathematical logic the Lowenheim number of an abstract logic is the smallest cardinal number for which a weak downward Lowenheim-Skolem theorem holds.[1] They are named after Leopold Lowenheim, who proved that these exist for a very broad class of logics. Examples ・The Lowenheim-Skolem theorem shows that the Lowenheim-Skolem-Tarski number of first-order logic is ?0. This means, in particular, that if a sentence of first-order logic is satisfiable, then the sentence is satisfiable in a countable model. ・It is known that the Lowenheim-Skolem number of second-order logic is larger than the first measurable cardinal, if there is a measurable cardinal.[3] (And the same holds for its Hanf number.) The Lowenheim number of the universal (fragment of) second-order logic however is less than the first supercompact cardinal (assuming it exists).
https://en.wikipedia.org/wiki/Measurable_cardinal Measurable cardinal In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal ｋ, or more generally on any set. For a cardinal ｋ, it can be described as a subdivision of all of its subsets into large and small sets such that ｋ itself is large, Φ and all singletons {α}, α ∈ ｋ are small, complements of small sets are large and vice versa. The intersection of fewer than ｋ large sets is again large.[1]
It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC.[2] The concept of a measurable cardinal was introduced by Stanislaw Ulam in 1930.[3]
Properties Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. (引用終り) 以上
下記（山上滋 名大）「（ノイマン 自然数）しかし、これは、落ち着いて考えてみると、Φ の記号を取り囲む括弧の数を数えているに過ぎないのであって、当然といえば当然のことである。」 同様下記（Axiom of infinity）" The count of elements in each set, at the top level, is the same as the represented natural number, and the nesting depth of the most deeply nested empty set {}, including its nesting in the set that represents the number of which it is a part,"
要するに、ノイマンの自然数の集合Ｎが出来上がったとき Ｎは無限集合だが、同時に「Φ の記号を取り囲む括弧の数」（山上滋） （あるいは”the nesting depth of the most deeply nested empty set {}”（Axiom of infinity）） は、可算無限である 下記（山上滋）「Φ, {Φ}, {{Φ}}, . . .」で、この列は有限であってはならない！ 可算無限である。よって、「Φ の記号を取り囲む括弧」の可算無限のシングルトンが存在する これは、無限公理から従う それを、{・・{Φ}・・}と表現するか、・・{Φ}・・、あるいは{・・Φ・・}か そんなことは、どうでも良いこと！ 幼稚な話にすぎない！（＾＾
https://en.wikipedia.org/wiki/Axiom_of_infinity Axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo?Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] Formal statement In the formal language of the Zermelo?Fraenkel axioms, the axiom reads: ∃ Ｉ (Φ ∈ Ｉ ∧ ∀x∈ Ｉ ((x∪{x})∈ Ｉ )). In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set. This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I. Interpretation and consequences This axiom is closely related to the von Neumann construction of the natural numbers in set theory, in which the successor of x is defined as x ∪ {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set: 0 = {}. The number 1 is the successor of 0: 1 = 0 ∪ {0} = {} ∪ {0} = {0} = {{}}. Likewise, 2 is the successor of 1: 2 = 1 ∪ {1} = {0} ∪ {1} = {0,1} = { {}, {{}} },
and so on: 3 = {0,1,2} = { {}, {{}}, {{}, {{}}} }; 4 = {0,1,2,3} = { {}, {{}}, { {}, {{}} }, { {}, {{}}, {{}, {{}}} } }. A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers.
The count of elements in each set, at the top level, is the same as the represented natural number, and the nesting depth of the most deeply nested empty set {}, including its nesting in the set that represents the number of which it is a part, is also equal to the natural number that the set represents.
Thus the essence of the axiom is: There is a set, I, that includes all the natural numbers. The axiom of infinity is also one of the von Neumann?Bernays?Godel axioms. Extracting the natural numbers from the infinite set The infinite set I is a superset of the natural numbers. To show that the natural numbers themselves constitute a set, the axiom schema of specification can be applied to remove unwanted elements, leaving the set N of all natural numbers. This set is unique by the axiom of extensionality.
To extract the natural numbers, we need a definition of which sets are natural numbers. The natural numbers can be defined in a way which does not assume any axioms except the axiom of extensionality and the axiom of induction?a natural number is either zero or a successor and each of its elements is either zero or a successor of another of its elements. In formal language, the definition says: 略 This definition is convenient because the principle of induction immediately follows: If I⊆ω is inductive, then also ω ⊆I, so that I=ω.
Both these methods produce systems which satisfy the axioms of second-order arithmetic, since the axiom of power set allows us to quantify over the power set of ω , as in second-order logic. Thus they both completely determine isomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal. (引用終り) 以上
>>668 追加 >Axiom of infinity >In the formal language of the Zermelo?Fraenkel axioms, the axiom reads: >∃ Ｉ (Φ ∈ Ｉ ∧ ∀x∈ Ｉ ((x∪{x})∈ Ｉ )). >In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. >Such a set is sometimes called an inductive set.
（参考） 突然ですが、Zermelo set theory がヒットしたので貼る（＾＾ https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo set theory Zermelo set theory (sometimes denoted by Z-), as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo-Fraenkel set theory (ZF) and its extensions, such as von Neumann?Bernays?Godel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. Contents 1 The axioms of Zermelo set theory 2 Connection with standard set theory 3 Mac Lane set theory 4 The aim of Zermelo's paper 5 The axiom of separation 6 Cantor's theorem
The axioms of Zermelo set theory AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element."
Connection with standard set theory The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal ω; the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of V_{ω} as a set nor of any rank of the cumulative hierarchy of sets with infinite index. Zermelo allowed for the existence of urelements that are not sets and contain no elements; these are now usually omitted from set theories.
Mac Lane set theory Mac Lane set theory, introduced by Mac Lane (1986), is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded. Mac Lane set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.
The aim of Zermelo's paper The introduction states that the very existence of the discipline of set theory "seems to be threatened by certain contradictions or "
749 名前：antinomies", that can be derived from its principles ? principles necessarily governing our thinking, it seems ? and to which no entirely satisfactory solution has yet been found". Zermelo is of course referring to the "Russell antinomy".
Cantor's theorem Zermelo's paper may be the first to mention the name "Cantor's theorem".
（ついで） https://en.wikipedia.org/wiki/S_(set_theory) S (set theory) S is an axiomatic set theory set out by George Boolos in his 1989 article, "Iteration Again". S, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. (引用終り) 以上 []
>>726 補足 (引用開始) https://researchmap.jp/multidatabases/multidatabase_contents/detail/227670/99650f5ed6c6962b8c9bfde71d7373b1?frame_id=640295 新井 敏康 東京大学数理科学研究科「数学基礎論・数理科学続論D」の講義資料 無限集合の存在は要請されなければならない： ω := N := {n : n is a natural number} は集合である(Axiom of Infinity) つまりω =∩{x : 0 ∈ x∧∀y ∈ x(y ∪{y} ∈ x)} で、かつ最小のlimit ordinal である。 (引用終り)
なるほど これは、オリジナルの無限公理よりも良いかも つまり、オリジナルの無限公理は、自然数の集合Ｎを含むもっと大きな集合が出来てしまう そこから、自然数の集合Ｎに絞り込むのに、二階述語論理を使うが（下記en.wikipedia）、できれば一階で済ませたい よって、新井流で、「ω := N := {n : n is a natural number} は集合である(Axiom of Infinity)」 とズバリ書いてしまう （ちょっと、記述が汚なくなりそうだけどね。”natural number”の定義をどう書くかの工夫がいりそうだし そもそも自然数の集合Ｎを定義するのに、”n is a natural number”とか、”最小のlimit ordinal”とか、そんなんあり？ みたいな でも、二階述語論理を使うのとどっちがどうか？ どうせ不完全性定理があるから、記述の汚さは多少妥協した新井流もありかな（＾＾ ） なるほどねぇ（＾＾；
参考 (>>668-670より) https://en.wikipedia.org/wiki/Axiom_of_infinity Axiom of infinity Extracting the natural numbers from the infinite set Both these methods produce systems which satisfy the axioms of second-order arithmetic, since the axiom of power set allows us to quantify over the power set of ω , as in second-order logic. Thus they both completely determine isomorphic systems, and since they are isomorphic under the identity map, they must in fact be equal.
だったら、新井流で「ω := N := {n : n is a natural number} は集合である(Axiom of Infinity)」 みたく 自然数の集合Ｎを公理として入れてしまえて、そして公理の外では一階述語論理で通せれば、それはそれで綺麗だろ 一方、ツェルメロやノイマンたちは、あくまで公理はシンプルであるべしと考えていた （公理に使う用語や概念は、極小にしたい。{}とか∪,∩,∀,∃とかね。余計な用語は公理には極力入れたくない） それで、通せると思っていた。当時の全数学を構築できると で、ゲーデルの不完全性定理が出て、結局公理は追加される運命にあると分かったんだ だったら、新井流で最初からNを公理中で与えるやり方もありかもと思った。多少公理が複雑になってもね（＾＾
（>>456-459より） http://www.cs-study.com/koga/set/lowenheimSkolem.html 形式的論理体系の定義から レーベンハイム・スコーレムの定理までの大急ぎのまとめ (Rapid Summary from Syntax of Logic to Lowenheim-Skolem Theorem) by Akihiko Koga 27th Mar. 2020 (Update) レーベンハイム・スコーレムの定理（レーベンハイム発表 1915年，スコーレムによる厳密な証明 1920年）は，一階の記号論理体系（一階述語論理）の「モデル（その体系の公理系を 満たす数学的な実例）」のサイズに関する定理である． レーベンハイム・スコーレムの定理は，このときの記号を解釈するための「実体の集合 M」の 大きさに関する命題である．より詳しく言うと， 記号論理の体系がモデルを持つと 分かったとき，そのモデルを非常に巨大な大きさにしたり，またはその逆に， 非常に小さくしたりできると いう定理である． http://www.cs-study.com/koga/set/pictures/Lowenheim00.png 一階述語論理でレーベンハイム・スコーレムの定理が成立するということは，一階述語論理では 無限集合の実際の大きさを論理式で限定できないことを意味する． (引用終り) 以上 []
https://en.wikipedia.org/wiki/Real_number Real number
Contents 1 History 2 Definition 2.1 Axiomatic approach 2.2 Construction from the rational numbers Axiomatic approach Let {\displaystyle \mathbb {R} }\mathbb {R} denote the set of all real numbers, then:
The set {\displaystyle \mathbb {R} }\mathbb {R} is a field, meaning that addition and multiplication are defined and have the usual properties. The order is Dedekind-complete, meaning that every non-empty subset S of {\displaystyle \mathbb {R} }\mathbb {R} with an upper bound in {\displaystyle \mathbb {R} }\mathbb {R} has a least upper bound (a.k.a., supremum) in {\displaystyle \mathbb {R} }\mathbb {R} . For another axiomatization of {\displaystyle \mathbb {R} }\mathbb {R} , see Tarski's axiomatization of the reals. Construction from the rational numbers The real numbers can be constructed as a completion of the rational numbers, in such a way that a sequence defined by a decimal or binary expansion like (3; 3.1; 3.14; 3.141; 3.1415; ...) converges to a unique real number?in this case π. For details and other constructions of real numbers, see construction of the real numbers.
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers Construction of the real numbers
Contents 1 Synthetic approach 1.1 Axioms 1.1.1 On the least upper bound property 1.1.2 On models 1.2 Tarski's axiomatization of the reals 2 Explicit constructions of models 2.1 Construction from Cauchy sequences 2.2 Construction by Dedekind cuts 2.3 Construction using hyperreal numbers 2.4 Construction from surreal numbers 2.5 Construction from integers (Eudoxus reals) 2.6 Other constructions (引用終り) 以上
https://en.wikipedia.org/wiki/Natural_number Natural number Constructions based on set theory Main article: Set-theoretic definition of natural numbers See also: Ordinal number § Definitions
>>767 Axiom of regularity が、帰納法の公理と関係しているそうだよ （下記）
（参考） https://en.wikipedia.org/wiki/Axiom_of_regularity Axiom of regularity Given the other axioms of Zermelo?Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. The axiom of induction tends to be used in place of the axiom of regularity in intuitionistic theories (ones that do not accept the law of the excluded middle), where the two axioms are not equivalent.
https://encyclopediaofmath.org/wiki/Induction_axiom Induction axiom The assertion of the validity for all x of some predicate P(x) defined on the set of all non-negative integers, if the following two conditions hold: 1) P(0) is valid; and 2) for any x, the truth of P(x) implies that of P(x+1).
The induction axiom is written in the form P(0)&∀x(P(x)⊃P(x+1))⊃∀x P(x). In applications of the induction axiom, P(x) is called the induction predicate, or the induction proposition, and x is called the induction variable,
This axiom is called the complete or recursive induction axiom. The principle of complete induction is equivalent to the principle of ordinary induction. See also Transfinite induction.
https://en.wikipedia.org/wiki/Epsilon-induction Epsilon-induction In mathematics, {\displaystyle \in }\in -induction (epsilon-induction or set-induction) is a variant of transfinite induction. Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction. It can be used in set theory to prove that all sets satisfy a given property P(x). This is a special case of well-founded induction. Contents 1 Statement 1.1 Comparison with natural number induction 2 Independence Comparison with natural number induction The above can be compared with {\displaystyle \omega }\omega -induction over the natural numbers {\displaystyle n\in \{0,1,2,\dots \}}{\displaystyle n\in \{0,1,2,\dots \}} for number properties Q. Independence In the context of the constructive set theory CZF, adopting the Axiom of regularity would imply the law of excluded middle and also set-induction. But then the resulti
855 名前：ng theory would be standard ZF. However, conversely, the set-induction implies neither of the two. In other words, with a constructive logic framework, set-induction as stated above is strictly weaker than regularity.
下記だな ”Thus, for every topological space X, the topos Sh(X) has a Dedekind real numbers object R. Naively one might expect R to be isomorphic to the constant sheaf Δ(R), where R is the classical set of real numbers, but this turns out not to be the case. Instead, we have a rather more remarkable result: Theorem 5.1.略”
（参考） https://ncatlab.org/nlab/show/real+numbers+object Real numbers object 1. Idea 2. Definition 3. Properties 4. Constructions In a topos with an NNO In a Π-pretopos with WCC In a Π-pretopos with an NNO and subset collection 5. Examples In Set In sheaves on a topological space In sheaves on a gros site of topological spaces In a general sheaf topos 6. Generalizations Cauchy real numbers Classical Dedekind real numbers 7. Related concepts
1. Idea Recall that it is possible to define an internalization of the set of natural numbers, called a natural numbers object (NNO), in any cartesian monoidal category (a category with finite products). In particular, the notion makes sense in a topos. But a topos supports intuitionistic higher-order logic, so once we have an NNO, it is also possible to repeat the usual construction of the integers, the rationals, and then finally the real numbers; we thus obtain an internalization of R in any topos with an NNO.
More generally, we can define a real numbers object (RNO) in any category with sufficient structure (somewhere between a cartesian monoidal category and a topos). Then we can prove that an RNO exists in any topos with an NNO (and in some other situations).
2. Definition Let E be a Heyting category. (This means, in particular, that we can interpret full first-order intuitionistic logic using the stack semantics.)
5. Examples In Set The real numbers object in Set is the real line, the usual set of (located Dedekind) real numbers. Note that this is a theorem of constructive mathematics, as long as we assume that Set is an elementary topos with an NNO (or more generally a Π-pretopos with NNO and either WCC or subset collection).
In sheaves on a topological space
Thus, for every topological space X, the topos Sh(X) has a Dedekind real numbers object R. Naively one might expect R to be isomorphic to the constant sheaf Δ(R), where R is the classical set of real numbers, but this turns out not to be the case. Instead, we have a rather more remarkable result:
Theorem 5.1. A Dedekind real numbers object R in the topos Sh(X) is isomorphic to the sheaf of real-valued continuous functions on X.
This is shown in (MacLane-Moerdijk, Chapter VI, §8, theorem 2); see also below.
Remark 5.2. Theorem 5.1 allows us to define various further constructions on X in internal terms in Sh(X); for example, a vector bundle over X is an internal projective R-module. (引用終り) 以上
それでね、>>768を補足しておくよ ”Given the other axioms of Zermelo-Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction.” とあるよね つまり、ノイマンが基礎の公理（the axiom of regularity）を導入した大きな意図が、ここにあるんだ
>>769から Epsilon-induction In mathematics, ∈-induction (epsilon-induction or set-induction) is a variant of transfinite induction. Considered as an alternative set theory axiom schema, it is called the Axiom (schema) of (set) induction. It can be used in set theory to prove that all sets satisfy a given property P(x). This is a special case of well-founded induction. (引用終り)
とあるよね。（なお、”∈”を、ε（Epsilon）と呼ぶってことな、念のため）
https://en.wikipedia.org/wiki/Axiom_of_regularity Axiom of regularity regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on {(n,α )｜ n∈ ω ∧ α is an ordinal }. (引用終り) ってこと
つまり、基礎の公理が、”not only allows induction”で、”but also on proper classes that are well-founded relational structures such as the lexicographical ordering on {(n,α )｜ n∈ ω ∧ α is an ordinal }” を意図しているってことです
>>789-790 では問う Q1. ノイマンの自然数構成で 0∈1∈2・・∈N（＝ω） なる”∈”による無限の上昇列ができると思うが、どうか？ Y or N？ まさか、これが有限列だとでも？ 基礎の公理に違反するとでも？ｗ（＾＾；
Q2.正則性公理にだけ反するというなら、正則性公理のない公理系の場合は存在しうるのでは？ （下記”Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980).”ご参照）
どぞ、ご回答を。怖気づいて回答できないかもね？（＾＾；
（参考） https://en.wikipedia.org/wiki/Axiom_of_regularity Axiom of regularity Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). Sources Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Elsevier, ISBN 978-0-444-86839-8
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument Cantor's diagonal argument In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.[1] References [1] https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002113910&physid=phys84#navi Georg Cantor (1891). "Ueber eine elementare Frage der Mannigfaltigkeitslehre". Jahresbericht der Deutschen Mathematiker-Vereinigung. 1: 75?78.
https://cs.maryvillecollege.edu/wiki/images/c/cb/Cantor_UeberEineElementare_Trans_v1.pdf A Translation of G. Cantor’s “Ueber eine elementare Frage der Mannigfaltigkeitslehre”. Google TranslateTM,1 DeepLTM,2 and Peter P. Jones? 1 https: // translate. google. com 2 https: // www. deepl. com (Dated: August 23, 2019) An English translation of G. Cantor’s “Ueber eine elementare Frage der Mannigfaltigkeitslehre”[1] article: “On an elementary question of the theory of manifolds.” Translation Note: We have translated “Inbegriff” as collection, and “M¨achtigkeit” as power. Apart from these adjustments and a few other specific edits the bulk of this English language text was obtained directly from the machine translators acknowledged as the main authors. (引用終り) 以上
https://ja.wikipedia.org/wiki/%E7%A8%A0%E5%AF%86%E9%9B%86%E5%90%88 位相空間 X の部分集合 A が X において稠密（ちゅうみつ、英: dense）であるとは、X の各点 x が、A の元であるか、さもなくば A の集積点であるときにいう[1]。イメージで言えば、X の各点が A の中か、さもなくば A の元の「どれほどでも近く」にあるということを表している。例えば、有理数は実数の稠密集合である。なぜなら任意の実数は、有理数であるか、さもなくばどれほどでも近い有理数をとることができるからである（ディオファントス近似も参照）。
https://ja.wikipedia.org/wiki/%E9%9B%86%E7%A9%8D%E7%82%B9 集積点（しゅうせきてん、英: accumulation point）あるいは極限点（きょくげんてん、英: limit point）は、位相空間 X の部分集合 S に対して定義される概念。 極限点の種類 ・x を含む任意の開集合が無限に多くの S の点を含むとき、集積点 x を特に S の ω-集積点 (ω-accumulation point) という。 ・x を含む任意の開集合が非可算無限個の S の点を含むとき、集積点 x を特に S の凝集点 (condensation point) という。 (引用終り) 以上
>>860 補足 追加の追加 （参考：英語版）（＾＾ https://en.wikipedia.org/wiki/Well-order Well-order Examples and counterexamples Reals The standard ordering ≦ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals. Also Wacław Sierpiński proved that ZF + GCH (the generalized continuum hypothesis) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.[1] However it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that V=L, and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.
An uncountable subset of the real numbers with the standard ordering ≦ cannot be a well order: Suppose X is a subset of R well ordered by ≦. For each x in X, let s(x) be the successor of x in ≦ ordering on X (unless x is the last element of X). Let A = { (x, s(x)) | x ∈ X } whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an injective function from A to Q. There is an injection from X to A (except possibly for a last element of X which could be mapped to zero later). And it is well known that there is an injection from Q to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from X to the natural numbers which means that X is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard "≦". For example,
・The natural numbers are a well order under the standard ordering ≦. ・The set {1/n : n =1,2,3,...} has no least element and is therefore not a well order under standard ordering ≦. Examples of well orders: ・The set of numbers { - 2^-n | 0 ≦ n < ω } has order type ω. ・The set of numbers { - 2^-n - 2-m-n | 0 ≦ m,n < ω } has order type ω2. The previous set is the set of limit points within the set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point of the set of limit points. ・The set of numbers { - 2^-n | 0 ≦ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 is a limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers it is not. (引用終り) 以上
>>875 補足 なんか、分かってないね （下記より） ・整列集合：集合 S 上の整列順序関係 (wellorder) とは、S 上の全順序関係 "≦" であって、S の空でない任意の部分集合が必ず ≦ に関する最小元をもつものをいう つまり、全順序に「必ず ≦ に関する最小元をもつ」という条件を加えたもの ・全順序：線型順序、元を直線に並べた図式によってその集合が表せるということでもあり、それは「線型」順序の名の由来である 「集合 X が関係 ≦ による全順序をもつとは、X の任意の元 a, b, c に対して、次の3条件を満たすことである」
”X の任意の元 a, b, c に対して”にご注目 任意とは、英語では anyかallですが、数学では∀ですよ！（＾＾ （参考） https://ja.wikipedia.org/wiki/%E6%95%B4%E5%88%97%E9%9B%86%E5%90%88 整列集合（well-ordered set）とは、整列順序を備えた集合のことをいう。ここで、集合 S 上の整列順序関係 (well-order) とは、S 上の全順序関係 "≦" であって、S の空でない任意の部分集合が必ず ≦ に関する最小元をもつものをいう。あるいは同じことだが、整列順序とは整礎な全順序関係のことである。整列集合 (S, ≦) を慣例に従ってしばしば単純に S で表す。
https://ja.wikipedia.org/wiki/%E5%85%A8%E9%A0%86%E5%BA%8F 全順序 単純順序（たんじゅんじゅんじょ、英: simple order）、線型順序（せんけいじゅんじょ、英: linear order）とも呼ばれる。 集合 X が関係 ≦ による全順序をもつとは、X の任意の元 a, b, c に対して、次の3条件を満たすことである： 反対称律：a ≦ b かつ b ≦ a ならば a = b 推移律：a ≦ b かつ b ≦ c ならば a ≦ c 完全律（比較可能）：a ≦ b または b ≦ a の何れかが必ず成り立つ 反対称性によって a < b かつ b < a であるという不確定な状態は排除される[1]。完全性を持つ関係は、その集合の任意の二元がその関係で比較可能（英語版）であることを意味する。これはまた、元を直線に並べた図式によってその集合が表せるということでもあり、それは「線型」順序の名の由来である[2]。また完全性から反射性 (a ≦ a) が出るから、全順序は半順序の公理を満たす。半順序は（完全性の代わりに反射性のみが課されるという意味で）全順序よりも弱い条件である。 (引用終り)
１．整列順序とは、全順序であって、任意の部分集合が極小元を持つ ２．従属選択公理（選択公理でも）を使えば、「関係が整礎であることを可算無限降下列が存在しないこととして定められる」 ３．全順序とは、「元を直線に並べた図式によってその集合が表せるということでもあり、それは「線型」順序の名の由来である」 ４．実数全体の成す集合 R は通常の大小関係 ("<" あるいは ">") によって全順序付けられる 従ってその部分集合としての、自然数全体の成す集合 N, 整数全体の成す集合 Z, 有理数全体の成す集合 Q なども全順序集合になる ５．当然、R は通常の大小関係 ("<" あるいは ">") によって、無限降下列も、あるいは無限上昇列も持つ なお、有理数全体の成す集合 Qは、可算無限に限られる ６．自然数全体の成す集合 Nは、整列順序であり、最小限を持ち、降下列は有限である ７．しかし、N∪ωを考えると（wiki/Well-founded_relationより）”Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n ? 1, n ? 2, ..., 2, 1 has length n for any n.” つまり、N∪ωでは、「∀降下列は、有限」が不成立（詳しくは下記英文嫁め）
（参考） https://ja.wikipedia.org/wiki/%E6%95%B4%E5%88%97%E9%9B%86%E5%90%88 整列順序付けられた集合または整列集合（せいれつしゅうごう、英: well-ordered set）とは、整列順序を備えた集合のことをいう。ここで、集合 S 上の整列順序関係 (well-order) とは、S 上の全順序関係 "≦" であって、S の空でない任意の部分集合が必ず ≦ に関する最小元をもつものをいう。あるいは同じことだが、整列順序とは整礎な全順序関係のことである。整列集合 (S, ≦) を慣例に従ってしばしば単純に S で表す。
https://ja.wikipedia.org/wiki/%E6%95%B4%E7%A4%8E%E9%96%A2%E4%BF%82 二項関係が整礎（well-founded）であるとは、真の無限降下列をもたないことである。 定義 集合あるいはクラス X 上の二項関係 R が整礎であるとは、X の空でない任意の部分集合 S が R に関する極小元を持つことをいう[1]。（関係 R がさらに集合的であることを仮定する著者もいる[2]。X が集合であればこれは自動的に成り立つ。）つまり、S の元 m であって、S の任意の元 s に対して対 (s, m) は R に属さないようなものが存在する。式で書けば ∀ S⊆ X(S≠Φ → ∃m∈ S ∀s∈S(s,m)not∈ R). X が集合であるとき、従属選択公理（英語版）（これは選択公理よりも真に弱く可算選択公理よりも真に強い）を仮定すれば、同値な定義として、関係が整礎であることを可算無限降下列が存在しないこととして定められる[3]。つまり、X の元の無限列 x0, x1, x2, ... で、どんな n についても xn+1 R xn となるようなものはとれない。
（上記「∀ S⊆ X(S≠Φ → ∃m∈ S ∀s∈S(s,m)not∈ R)」関連は英文で分かり易く加筆されているね ） https://en.wikipedia.org/wiki/Well-founded_relation In mathematics, a binary relation R is called well-founded (or wellfounded) on a class X if every non-empty subset S ⊆ X has a minimal element with respect to R, that is, an element m not related by sRm (for instance, "s is not smaller than m") for any s ∈ S.
In other words, a relation is well founded if (∀S⊆ X)[S≠ Φ ⇒ (∃m∈ S)(∀s∈ S)￢(sRm)].
Other properties If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n ? 1, n ? 2, ..., 2, 1 has length n for any n. The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X, R) is isomorphic to (C, ∈).
（>>871より再録） https://ja.wikipedia.org/wiki/%E5%85%A8%E9%A0%86%E5%BA%8F 全順序（ぜんじゅんじょ、英: total order）とは、集合での二項関係で、推移律、反対称律かつ完全律の全てを満たすもののことである。 元を直線に並べた図式によってその集合が表せるということでもあり、それは「線型」順序の名の由来である[2]。 例 実数全体の成す集合 R は通常の大小関係 ("<" あるいは ">") によって全順序付けられる。従ってその部分集合としての、自然数全体の成す集合 N, 整数全体の成す集合 Z, 有理数全体の成す集合 Q なども全順序集合になる。これらは何れも、ある性質に関して最小の全順序集合として（同型を除いて）唯一の例を与えることが示せる（ここで、全順序集合 A がある性質に関して「最小」とは、同じ性質を持つ任意の B に対して A に順序同型な B の部分集合が存在することをいう）。 ・N は上界を持たない最小の全順序集合である。 ・Z は上界も下界も持たない最小の全順序集合である。 ・Q は R の中で稠密となる最小の全順序集合である。ここでいう稠密性は a < b なる任意の実数 a, b に対し、a < q < b となる有理数 q が必ず存在することを言う。 ・R は順序位相（後述）に関して連結となる最小の非有界全順序集合である。 ・順序体は定義により全順序である。これは有理数体 Q や実数体 R を包括する概念である。 (引用終り) 以上
>>888 > Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. > Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n ? 1, n ? 2, ..., 2, 1 has length n for any n.
翻訳は以下のとおり
「次のような例を考えてみましょう。Xを正の整数と、任意の整数よりも大きい新要素ωとの和とする。 このとき、Xはwell-foundedな集合であるが、ωから始まる任意の大きな（有限の）長さの下降鎖があり、その鎖はω, n ? 1, n ? 2, ..., 2, 1 は任意の n に対して長さ n を持つ。」
（参考） https://www.アマゾン/product-reviews/4535781613 無限からの光芒―ポーランド学派の数学者たち 1988 by 志賀 浩二 日本評論社 Customer reviews まげ店長 5.0 out of 5 stars ポーランドにおける無限を巡る数学の発展について Reviewed in Japan on November 2, 2013 ポーランド史、無限論（カントール）、ユダヤ人史に興味の有る方にはうってつけの本です。 私はたまたまワルシャワ蜂起と集合論と数学基礎論を勉強しているので、この条件には見事に当てはまるのです。
P15 小平邦彦氏はこの対角線論法を評して、「簡単明瞭であるが、何かうまく言いくるめられた感じがしないでもない。そこで別証を考えて見る。(「数学の学び方」 [71]より)」とされ、測度論的証明を自著『解析入門』(72)に書き記されている。更に、「これで実数全体の集合 R が非可算であることの別証が得られたのである。別証は対角線論法による証明よりも面倒であるが、うまくいいくるめられたという感じはない。別証により R は非可算であるだけでなく、R の可算部分集合は Rの極めて小さい部を占めるに過ぎないことがわかる。(同上)」と、してこの別証明53の持つ意義を力説されている。 (引用終り)
定義4.2.1 (順序関係). 集合 A 上の関係 ≦ が順序関係、または単に順序であるとは、以下の条件を満たすこととする。
[反射律] 任意の x ∈ A に対して x ≦ x [推移律] x ≦ y , y ≦ z ならば x ≦ z [非対称律] x ≦ y , y ≦ x ならば x = y このとき (A, ≦) を順序集合という。
定義4.2.4 (全順序). 順序集合 (A, ≦ ) の任意の二つの要素 x,y ∈ A に対して x ≦ y または y ≦ x が成り立つとき、この順序を全順序といい、この順序集合を全順序集合という。
例4.2.6. 前述の「小学生を背の低い順に並べる」ということを考えよう。ある小学校のクラスの生徒を、ある身体測定の際の身長の小さい順に並べるとする。より一般に、集合 X と写像 f : X → R が与えられ、 f による値によって、集合 X の順序を決めるということを考えよう。
順序集合 (A,≦ ) の元 x に対して x ≦ y ならば x = y が成り立つとき x を A の極大元という。同様に y ≦ x ならば x = y であるとき x を A の極小元という。任意の y ∈ A に対して y ≦ x のとき x を A の最大元という。任意の y ∈ A に対して x ≦ y のとき x を A の最小元という。最大元は極大元、最小元は極小元であるが、逆は成り立つとは限らない。最大元、最小元は存在するとは限らないが、存在すれば唯一つに定まる。 (引用終り) 以上
例1.1.4. (￢A ) =⇒ A は常に偽であるように思われるが、先に述べたようにA が真であればこれは真である。 注意. A が偽であれば、任意の命題B に対してA =⇒ B は真になるのだが、これが感覚的に受け入れがたいという学生も少なくはない。先の説明ではA =⇒ B を感覚的に理解したが、正確にはA = ⇒ B の定義が(￢A ) ∨ B であり、認めてもらうしかない。
上の 7 による A → B の真理表は，次の ¬A ∨ B の真理表と結果が同じになることが分かる． A B ¬A ¬A ∨ B t t f t t f f f f t t t f f t t この意味で，→ は ¬ と ∨ を使って A → B を ¬A ∨ B の形で定義可能であることが分かる． (引用終り) 以上
