純粋・応用数学（含むガロア理論）8

**１３２人目の素数さん** · 2021/05/13(木) 20:12:42.63

クレレ誌：
https://ja.wikipedia.org/wiki/%E3%82%AF%E3%83%AC%E3%83%AC%E8%AA%8C
クレレ誌はアカデミーの紀要ではない最初の主要な数学学術誌の一つである（Neuenschwander 1994, p. 1533）。ニールス・アーベル、ゲオルク・カントール、ゴットホルト・アイゼンシュタインらの研究を含む著名な論文を掲載してきた。
(引用終り)

そこで
現代の純粋・応用数学（含むガロア理論）を目指して
新スレを立てる（＾＾；

＜前スレ＞
純粋・応用数学（含むガロア理論）7
https://rio2016.5ch.net/test/read.cgi/math/1618711564/

＜関連姉妹スレ＞
ガロア第一論文及びその関連の資料スレ
https://rio2016.5ch.net/test/read.cgi/math/1615510393/1-
箱入り無数目を語る部屋
https://rio2016.5ch.net/test/read.cgi/math/1609427846/
Inter-universal geometry と ABC予想 (応援スレ) 54
https://rio2016.5ch.net/test/read.cgi/math/1617170015/
IUTを読むための用語集資料スレ2
https://rio2016.5ch.net/test/read.cgi/math/1606813903/
現代数学の系譜カントル超限集合論他 3
https://rio2016.5ch.net/test/read.cgi/math/1595034113/

＜過去スレの関連（含むガロア理論）＞
・現代数学の系譜工学物理雑談古典ガロア理論も読む84
https://rio2016.5ch.net/test/read.cgi/math/1582200067/
・現代数学の系譜工学物理雑談古典ガロア理論も読む83
https://rio2016.5ch.net/test/read.cgi/math/1581243504/

**現代数学の系譜雑談** ◆yH25M02vWFhP · 2021/05/27(木) 22:28:22.15

>>521

HOL ”METAPHYSICS”だよ～ん。 HOLは、数学独占じゃない！！（＾＾；
http://tedsider.org/teaching/higher_order_20/higher_order_20.html
SEMINAR ON HIGHER ORDER METAPHYSICS
Rutgers Philosophy Department, 106 Somerset St, 5th floor, Fridays, 9:50-12:50, Spring 2020
Ted Sider, Room 526, office hours TBA and by appointment

Syllabus
Handout: philosophy of logic and second-order logic http://tedsider.org/teaching/higher_order_20/HO_CC_second_order_logic.pdf
Handout: paradoxes and set theory http://tedsider.org/teaching/higher_order_20/HO_crash_course_paradoxes.pdf
Handout: type theory and lambda abstraction http://tedsider.org/teaching/higher_order_20/HO_CC_type_theory_lambda_abstraction.pdf
Handout: Boolos http://tedsider.org/teaching/higher_order_20/HO_Boolos.pdf
Handout: Prior http://tedsider.org/teaching/higher_order_20/HO_Prior.pdf
Handout: Rayo and Yablo http://tedsider.org/teaching/higher_order_20/HO_Rayo_Yablo.pdf

http://tedsider.org/teaching/higher_order_20/higher_order_crash_course.pdf
Crash course on higher-order logic*
Theodore Sider August 14, 2020

Contents
1 Introduction 2
2 Importance of syntax to logic 3
2.1 Syntax in formal languages ..5
3 First- versus second-order logic 7
3.1 Syntax ... 7
3.2 Formal logic and logical consequence ..9
3.3 Semantics ...10
3.4 Proof theory ... 12
3.5 Metalogic ...16
3.5.1 Completeness ..16
3.5.2 Compactness ..17

3.6 Metamathematics ... 20
3.6.1 Skolem’s paradox ..21
3.6.2 Nonstandard models of arithmetic .21
3.6.3 Schematic and nonschematic axiomatizations .23
4 Paradoxes 26
4.1 Abstract mathematics and set-theoretic foundations . 26
4.2 Russell’s paradox ...28
4.3 Axiomatic set theory and ZF .. 30
4.4 Other paradoxes, other solutions ..34

5.1 Third-order logic and beyond ..37
5.2 Higher-order logic and types .. 38
略
以上

**現代数学の系譜雑談** ◆yH25M02vWFhP · 2021/05/27(木) 22:30:27.94

>>546
＞潔癖なだけでは？

何を言っているんだ
見る目がないね
潔癖だ？
こいつは、腐った魚以下
ただのサル
けものだよ

**現代数学の系譜雑談** ◆yH25M02vWFhP · 2021/05/28(金) 08:18:08.40

>>556 追加

higher-order logic topos で検索すると
高名な下記のSteve Awodey先生がヒット
Kohei Kishida Who?
” sheaf semantics, models are built on presheaves ”
なるほど、層や圏から、higher-order logicへ繋がっていくのか（＾＾

参考
https://arxiv.org/pdf/1403.0020.pdf
Topos Semantics for Higher-Order Modal Logic March 4, 2014
Steve Awodey? Kohei Kishida† Hans-Christoph Kotzsch‡
†Department of Computer Science, University of Oxford
(抜粋)
Abstract. We define the notion of a model of higher-order modal logic
in an arbitrary elementary topos E. In contrast to the well-known interpretation of (non-modal) higher-order logic, the type of propositions is not interpreted by the subobject classifier ΩE , but rather by a suitable complete Heyting algebra H.
The canonical map relating H and ΩE both serves to interpret equality and provides a modal operator on H in
the form of a comonad. Examples of such structures arise from surjective geometric morphisms f : F → E, where H = f?ΩF .
The logic differs from non-modal higher-order logic in that the principles of functional and propositional extensionality are not longer valid but may be replaced by modalized versions.
The usual Kripke, neighborhood, and sheaf semantics for propositional and first-order modal logic are subsumed by this notion.

In many conventional systems of semantics for quantified modal logic, models are built on presheaves.
Given a set K of “possible worlds”, Kripke’s semantics [11], for
instance, assigns to each world k ∈ K a domain of quantification P(k) - regarded
as the set of possible individuals that “exist” in k - and then ∃x Φ is true at k iff
some a ∈ P(k) satisfies Φ at k.
(引用終り)