現代数学の系譜工学物理雑談古典ガロア理論も読む62

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む62 (1002ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

205: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/03/13(水) 07:22:57.89 ID:QlfKIGCF

>>198
>>私は集合と要素を別のものとして区分するのは反対です
>無制限に、なんでも集合に取り入れると、まずいので公理化した
>で、素朴集合論から、公理的集合論（主としてZFC）の時代になった

（>>58）
http://blacaman.tripod.com/cursos/pdf/2012-2_0941.pdf
An Introduction to Independence Proofs K KUNEN 著 First edition: 1980 Seventh impression: 1999

https://www.amazon.co.jp/dp/4535783829/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=8NKTZE2Q63MR3BRQEWQX
集合論―独立性証明への案内 単行本 ? 2008/1/1
ケネス キューネン (著), Kenneth Kunen (原著), 藤田 博司 (翻訳)

下記が参考になるでしょう。
これの
P8
In the intended interpretation, under which the axioms of ZFC are presumed
true, x ∈ y is interpreted to mean that x is a member of y, but the
domain of discourse is somewhat harder to describe. In accordance with the
belief that set theory is the foundation of mathematics, we should be able
to capture all of mathematics by just talking about sets, so our variables
should not range over objects like cows and pigs. But if C is a cow, {C} is
a set, but not a legitimate mathematical object. More generally, since we
wish to talk only about sets but also should be able to talk about any element
of a set in our domain of discourse, all the elements of such a set should
be sets also.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1551963737/205

206: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/03/13(水) 07:24:46.64 ID:QlfKIGCF

>>205

つづき

P94
CHAPTER III THE WELL-FOUNDED SETS
§ 1. Introduction
Some questions about sets are irrelevant to mathematics.
First irrelevant question. Is there anything which is not a set? Certainly
there is in the "real world" of cows and pigs, but our axioms of set theory
say nothing about this "real world", since we have declared that they talk
only about sets - in fact, hereditary sets (see I §4). Furthermore, the Axiom
of Extensionality has embodied in it the assertion that all things in our
domain of discourse are sets. It seems likely that we have not left any interesting
mathematics behind by so restricting our universe, since mathematical
objects like R and C are hereditary sets and have been defined
explicitly within this domain in I § 11.
(引用終り)

つまり、ZFCでは、現実世界の素朴集合論の「 "real world" of cows and pigs」は、対象外だと
(部分引用)
”Is there anything which is not a set? Certainly
there is in the "real world" of cows and pigs, but our axioms of set theory
say nothing about this "real world", since we have declared that they talk
only about sets - in fact, hereditary sets (see I §4). ”
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1551963737/206

316: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/03/16(土) 09:52:59.90 ID:B5CZ4/Lr

>>205 補足（キューネン読んで）
（>>58）
http://blacaman.tripod.com/cursos/pdf/2012-2_0941.pdf
An Introduction to Independence Proofs K KUNEN 著 First edition: 1980 Seventh impression: 1999
https://www.amazon.co.jp/dp/4535783829/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=8NKTZE2Q63MR3BRQEWQX
集合論―独立性証明への案内 単行本 ? 2008/1/1
ケネス キューネン (著), Kenneth Kunen (原著), 藤田 博司 (翻訳)

P100
§4. The Axiom of Foundation
AXIOM 2. Foundation.
∀x(∃y(y∈x)→∃y(y∈x ∧ ￢∃z(z∈x ∧ z∈y))).

Equivalently, if x ≠ 0, ∃y ∈ X (x ∩ y = 0), or every non-empty set has
an ∈-minimal element, or, if we extend the definition of well-founded to
proper classes (see §5), ∈ is well-founded on V.
(引用終り)

”if x ≠ 0, ∃y ∈ X (x ∩ y = 0)”について
http://www.math.is.tohoku.ac.jp/~obata/student/subject/
東北大学大学院情報科学研究科 システム情報科学専攻 尾畑研究室－システム情報数理学II研究室－
尾畑伸明：集合・写像・数の体系　数学リテラシーとして
http://www.math.is.tohoku.ac.jp/~obata/student/subject/file/2018-3_shugo-enzan.pdf
第3章 集合の演算 GAIRON-book : 2018/6/21(19:23)
P34
x ∈ A ∩ B ←→ (x ∈ A) ∧ (x ∈ B) (3.2)
(引用終り)

なので
z∈x ∧ z∈y → z ∈ x ∩ y
￢∃z(z∈x ∧ z∈y) → z＝0 つまり x ∩ y = 0
よって
AXIOM 2. Foundation.
∀x(∃y(y∈x)→∃y(y∈x ∧ ￢∃z(z∈x ∧ z∈y))).
　↓
if x ≠ 0, ∃y ∈ X (x ∩ y = 0)
こっちの表現で記されているものが多い
多分、上の”AXIOM 2. Foundation”では、x,y,zと３つ出てくるが、こっちの表現だとx,yの２つで、よりシンプルってことだろうね
でも、x,y,zと３つ使う表現も、それはそれで意味あるんだろうね（キューネン先生が書いているんだから（＾＾ ）

つづく

http://rio2016.5ch.net/test/read.cgi/math/1551963737/316

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.039s