現代数学の系譜　カントル　超限集合論

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472: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/10/27(日) 20:54:28.82 ID:EUeYkluT

ほいよ

https://scholarpublishing.org/sse/
Services for Science and Education Ltd
https://scholarpublishing.org/sse/wp-content/uploads/2018/08/10.14738tnc.092018.1_global-set-theory_2018.pdf
Satoko Titani
Global Set Theory
Society for Science and Education (United Kingdom)

Dedicated to Professor Gaisi Takeuti (1925 ? 2017)

Contents
1 Basic set theory 11
1.1 Naive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Formal system of set theory . . . . . . . . . . . . . . . . . . . 16
1.2.1 Gentzen’s formal system of logic . . . . . . . . . . . . . 17
1.2.2 Inference rules of LK and LJ . . . . . . . . . . . . . . . 18
1.2.3 Axioms of set theory . . . . . . . . . . . . . . . . . . . 22
1.3 Construction of mathematics in ZFC . . . . . . . . . . . . . . 24
1.3.1 Definition of sets . . . . . . . . . . . . . . . . . . . . . 24
1.3.2 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3.5 Equivalence relation . . . . . . . . . . . . . . . . . . . 26
1.3.6 Natural numbers . . . . . . . . . . . . . . . . . . . . . 27
1.3.7 Operations on the natural numbers . . . . . . . . . . . 32
1.3.8 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.3.9 Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.3.10 Rational number . . . . . . . . . . . . . . . . . . . . . 42
1.3.11 Real number . . . . . . . . . . . . . . . . . . . . . . . . 43
1.3.12 Complex number . . . . . . . . . . . . . . . . . . . . . 44
1.3.13 Universe of ZFC . . . . . . . . . . . . . . . . . . . . . . 44

http://rio2016.5ch.net/test/read.cgi/math/1570237031/472

473: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/10/27(日) 23:44:36.69 ID:EUeYkluT

>>472 補足

https://scholarpublishing.org/sse/global-set-theory/
Services for Science and Education Ltd
Global Set Theory
Satoko Titani
Professor Emeritus of Chubu University, Japan.

Our reasoning is based on dichotomous logic. That is, we are naturally convinced that a statement is either true or false exclusively. We accept the dichotomy as axiom.
The dichotomous logic is a language of mathematics in which science is described. Set theory, which is the base of mathematics, is formulated into a formal system of a logic with set theoretical axioms. A logic provided with globalization or basic implication is called a global logic.
A set theory based on the global logic is called a global set theory. Global set theory comprehends the meta-theory of set theory.
By introducing the globalization into a set theory, we can express the truth value set and also express the universe of the set theory in the set theory. That is, global set theory is nested in the global set theory.
It follows that we can prove the completeness of global set theories such as lattice valued set theory and quantum set theory.
Logical science is founded on the base of global classical logic. Each classical statement is either true or false, and these outcomes are mutually exclusive. Thereby, the global classical theory determines true-or-false definitely.
The logical science gives us very fruitful information, even though it covers only a bounded aspect of nature that falls within the realm of logic. We see that our logic is not absolute and is in fact determined by the establishment of a truth value set, which depends on object world.
Nature in its entirety is far beyond the scope of logic. Therefore, logical science is unable to describe the entire natural universe, in the same way as a net is unable to scoop up everything in its path.

DOI: 10.14738/tnc.092018.1
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476: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/26(火) 07:52:14.65 ID:oYs7jyeH

現代数学の系譜 工学物理雑談 古典ガロア理論も読む79 より
https://rio2016.5ch.net/test/read.cgi/math/1573769803/169-
169 自分：現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 投稿日：2019/11/26(火) 00:26:15.90 ID:oYs7jyeH [3/5]
(抜粋)
シングルトンの可算多重カッコ（ {{{・・・{{{ }}}・・・}}} ←{ }が多重になったもの)
が理解できない落ちこぼれさんたち多数居たなｗｗ（＾＾；
(引用終り)

英文法では、数と序数詞が区別されるんだ
日本語では、助数詞で「‐番目」「-回目」「-人目」「‐位（順位）」を使うだよね

で、本題だが

数　　　　：1 　,2 　 ,3　　 ,4　　　 ,・・・,n　　　　　　,・・・∞
順序数　：1st,2nd ,3rd　 ,4th　　 ,・・・,nth　　 　　,・・・ω
（１対１対応）　↓↑
ｼﾝｸﾞﾙﾄﾝ：{}　,{{}}　,{{{}}}　 ,{{{{}}}}　　,・・・,{・・{}・・}n　,・・・{・・{}・・}∞

（注：{・・{}・・}nは、カッコ{}がn重、{・・{}・・}∞は、同∞重のｼﾝｸﾞﾙﾄﾝ）

ちゃんと、可算の範囲で、全部対応が付きますがなｗ（＾＾；

https://ja.wikipedia.org/wiki/%E5%BA%8F%E6%95%B0%E8%A9%9E
序数詞
(抜粋)
序数詞（じょすうし）、順序数詞（じゅんじょすうし）とは物事の順序・順番（序数）を表す数詞である。これに対し、物事の数量を表す数詞は基数詞と呼ばれる。
同音の助数詞との混同に注意。
欧州の言語において序数詞は、日付（日）や世紀、分数の分母、また1世、2世、3世…といった同名の人物の世代数などにも用いられる。

2.3 序数詞の発達していない言語
2.3.1 中国語
2.3.2 日本語
2.3.3 朝鮮語

日本語
日本語は単独の序数詞を持たず、「‐番目」「-回目」「-人目」「‐位（順位）」といった接尾辞や、「第‐」といった接頭辞を付けて順番・順序などの序数を表現する。

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/476

504: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/27(水) 22:09:56.09 ID:qnEhNItW

>>491
＞基礎付け問題

これは、下記が、元記事だな（＾＾

https://en.wikipedia.org/wiki/Finite_set
Finite set
(抜粋)
Contents
1 Definition and terminology
2 Basic properties
3 Necessary and sufficient conditions for finiteness
4 Foundational issues
5 Set-theoretic definitions of finiteness
5.1 Other concepts of finiteness

Foundational issues
Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite and the infinite lies at the core of set theory.
Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus recommend a mathematics based solely on finite sets.
Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo?Fraenkel set theory with the axiom of infinity replaced by its negation.

Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter.
The difficulty stems from Godel's incompleteness theorems. One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets.
In particular, there exists a plethora of so-called non-standard models of both theories. A seeming paradox is that there are non-standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/504

505: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/27(水) 22:11:18.53 ID:qnEhNItW

>>504

つづき

(This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.)
On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately.

More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC),
Von Neumann?Bernays?Godel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.

A formalist might see the meaning[citation needed] of set varying from system to system. Some kinds of Platonists might view particular formal systems as approximating an underlying reality.

Set-theoretic definitions of finiteness
In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form {\displaystyle \{x\,|\,x<n\}}{\displaystyle \{x\,|\,x<n\}}.
Mathematicians more typically choose to ground notions of number in set theory, for example they might model natural numbers by the order types of finite well-ordered sets. Such an approach requires a structural definition of finiteness that does not depend on natural numbers.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/505

506: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/27(水) 22:12:50.56 ID:qnEhNItW

>>505

つづき

Various properties that single out the finite sets among all sets in the theory ZFC turn out logically inequivalent in weaker systems such as ZF or intuitionistic set theories. Two definitions feature prominently in the literature, one due to Richard Dedekind, the other to Kazimierz Kuratowski. (Kuratowski's is the definition used above.)

A set S is called Dedekind infinite if there exists an injective, non-surjective function {\displaystyle f:S\rightarrow S}f:S\rightarrow S.
Such a function exhibits a bijection between S and a proper subset of S, namely the image of f. Given a Dedekind infinite set S, a function f, and an element x that is not in the image of f, we can form an infinite sequence of distinct elements of S, namely {\displaystyle x,f(x),f(f(x)),...}x,f(x),f(f(x)),....
Conversely, given a sequence in S consisting of distinct elements {\displaystyle x_{1},x_{2},x_{3},...}x_{1},x_{2},x_{3},..., we can define a function f such that on elements in the sequence {\displaystyle f(x_{i})=x_{i+1}}{\displaystyle f(x_{i})=x_{i+1}} and f behaves like the identity function otherwise.
Thus Dedekind infinite sets contain subsets that correspond bijectively with the natural numbers. Dedekind finite naturally means that every injective self-map is also surjective.

Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semilattice. Writing K(S) for the sub-semilattice generated by the empty set and the singletons, call set S
Kuratowski finite if S itself belongs to K(S).[8] Intuitively, K(S) consists of the finite subsets of S.
Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semilattices containing the empty set and the singletons.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/506

507: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/27(水) 22:13:33.06 ID:qnEhNItW

>>506
つづき

Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation. Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that:

・X contains the empty set;
・For every set T in P(S), if X contains T then X also contains the union of T with any singleton.

Then K(S) may be defined as the intersection of M.

In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs.
That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence. However, Kuratowski finiteness would fail for the same set of socks.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1570237031/507

509: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/28(木) 00:24:40.66 ID:QdpmOFrx

>>508

つづき

Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the powerset P(S) with the structure of a semilattice.
Writing K(S) for the sub-semilattice generated by the empty set and the singletons, call set S Kuratowski finite if S itself belongs to K(S).[8] Intuitively,
K(S) consists of the finite subsets of S. Crucially, one does not need induction, recursion or a definition of natural numbers to define generated by since one may obtain K(S) simply by taking the intersection of all sub-semilattices containing the empty set and the singletons.

Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation.
Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that:

X contains the empty set;
For every set T in P(S), if X contains T then X also contains the union of T with any singleton.
Then K(S) may be defined as the intersection of M.

In ZF, Kuratowski finite implies Dedekind finite, but not vice versa.
In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs.
That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence.
However, Kuratowski finiteness would fail for the same set of socks.

http://rio2016.5ch.net/test/read.cgi/math/1570237031/509

519: 現代数学の系譜 雑談 ◆e.a0E5TtKE  [] 2019/11/28(木) 21:01:57.22 ID:QdpmOFrx

>>501-502 補足
(引用開始)
天才Zermeloが、シングルトンによる自然数の構成を与えた（1908年）
（”The natural numbers are represented by Zermelo as by Φ, {Φ}, {{Φ}}, …, and the Axiom of Infinity gives us a set of these.”）
https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B0
自然数
例えば、0 := {}, suc(a) := {a} と定義したならば、
0 := {}
1 := {0} = {{}}
2 := {1} = {{{}}}
3 := {2} = {{{{}}}}
と非常に単純な自然数になる。
(引用終り)

さて
・上記のように、シングルトンは、有限には限らない
　（これは自明だが以下説明する）
・数学では、可算無限を考えることは、頻繁にある
・例えば、下記の時枝記事は”可算無限個ある.箱それぞれに，私が実数を入れる”という記載から始まる
・あるいは、下記の形式的冪級数の各項の係数が、”箱が可算無限個ある”ことに相当するだろう
・また、下記のヒルベルトの無限ホテルのパラドックスでは、”客室が無限にあるホテルを考える”となる
・さて、可算無限個ある箱に、縦棒”|”を入れるとする。”|||・・・”となる
　これを、利用して、・・・|||Φ|||・・・、
　つまりΦを真ん中にして、左右に”|||・・・”を配置する
・ここで、縦棒”|”を左カッコ{ や、右カッコ ｝に取り替える。即ち
　左の・・・|||→・・・{{{ に
　右の|||・・・→{{{・・・ に 取り替えると
　・・・{{{Φ}}}・・・となる
　ここで、Φを取り除けば、・・・{{{ }}}・・・
　ここでΦ＝{ }を替えれば、・・・{{{{ }}}}・・・となる
・ヒルベルトの無限ホテルや形式的冪級数の存在が、否定できない（当然できないよね）
　とすれば、”|||・・・”の存在も否定できない
・従って、・・・{{{ }}}・・・（可算無限多重シングルトン）の存在も否定できない
QED

つづく

http://rio2016.5ch.net/test/read.cgi/math/1570237031/519

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