課程名稱 |
基本集合論 Elementary Set Theory |
開課學期 |
107-1 |
授課對象 |
文學院 哲學研究所 |
授課教師 |
楊金穆 |
課號 |
Phl7713 |
課程識別碼 |
124EM2990 |
班次 |
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學分 |
3.0 |
全/半年 |
半年 |
必/選修 |
選修 |
上課時間 |
星期五7,8,9(14:20~17:20) |
上課地點 |
哲研討室二 |
備註 |
本課程以英語授課。C領域。 總人數上限:15人 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
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課程概述 |
The course is designed for philosophy undergraduates specially interested in formal logic, philosophy of mathematics and philosophy of language and logic. The aim of this course is to provide a nodding acquaintance with elementary properties of sets and basic axioms of set theory. The study will be confined to the well-established Zermelo -Fraenkel set theory. (A course on Elementary Logic is usually assumed, though not so necessary)
Content
Preliminary
.1 Some naive assumptions of set theory
.2 Two aspects of set theory and the development of set theory: A historical survey
.3 The mathematical and philosophical significance of set theory
.4 A first-order language suitable for set theory
1 Basic Axioms for Sets
2 Functions
3 Relations
4. Natural Numbers
4.1 The axiom of infinity and the construction of - the set of natural numbers
4.2 Peano's postulates
4.3 Recursion theorems on and arithmetic of
4.4 Orderings on
4.5 Peano Systems and Extensions of natural numbers
5 Equinumerous sets and Schroder-Bernstein Theorem
6 Cardinal Numbers
6.1 The need for cardinality
6.2 Finite and infinite sets
6.3 The axiom schema of replacement
7 The Axiom of Choice and Countable Sets
7.1 The axiom of choice
7.2 Countable sets
8 The Arithmetic of Cardinal Numbers
8.1 Cardinal addition
8.2 Cardinal multiplication (product)
8.3 Cardinal exponentiation
9 Zorn's Lemma, Continuum Hypothesis and Some Variants of the Axiom of Choice
9.1 Zorn's Lemma
9.2 Continuum Hypothesis
9.3 Some equivalent variants of Zorn's Lemma
10 Well-orderings
10.1 Well-orderings of sets
10.2 The comparison theorem for well-ordered sets
10.3 Well -ordering Principle
11. Ordinals
11.1 The Notion of Ordinal Numbers
11.2 Some elementary properties of the ordinals
11.3 Ordinal numbers and cardinal numbers
12 Arithmetic of Ordinals
Appendix. Models of ZF- Set Theory
@. 1 Natural models of ZF - set theory
@.2 The class of infinite c |
課程目標 |
The aim of this course is to provide a nodding acquaintance with elementary properties of sets and basic axioms of set theory. |
課程要求 |
待補 |
預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
H.B. Enderton, Elements of Set Theory, New York: Academic Press, 1977.
K. Kunen, Set Theory - An Introduction to Independence Proofs, Amsterdam: North-Holland, 1980. |
參考書目 |
H.B. Enderton, Elements of Set Theory, New York: Academic Press, 1977.
K. Kunen, Set Theory - An Introduction to Independence Proofs, Amsterdam: North-Holland, 1980.
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評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
Class performance |
20% |
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2. |
Weekly assignment |
30% |
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3. |
End-of-term exam |
50% |
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週次 |
日期 |
單元主題 |
第1週 |
9/14 |
Preliminary |
第2週 |
9/21 |
Basic Axioms for Sets |
第3週 |
9/28 |
Functions |
第4週 |
10/05 |
Relations |
第5週 |
10/12 |
Natural Numbers |
第6週 |
10/19 |
Equinumerous sets and Schröder-Bernstein Theorem |
第7週 |
10/26 |
Cardinal Numbers |
第8週 |
11/02 |
Cardinal Numbers |
第9週 |
11/09 |
The Axiom of Choice and Countable Sets
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第10週 |
11/16 |
The Arithmetic of Cardinal Numbers
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第11週 |
11/23 |
The Arithmetic of Cardinal Numbers
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第12週 |
11/30 |
Zorn's Lemma, Continuum Hypothesis and Some Variants of the Axiom of Choice
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第13週 |
12/07 |
Zorn's Lemma, Continuum Hypothesis and Some Variants of the Axiom of Choice
Well-orderings
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第14週 |
12/14 |
Ordinals |
第15週 |
12/21 |
Ordinals |
第16週 |
12/28 |
Arithmetic of Ordinals
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第17週 |
1/04 |
Arithmetic of Ordinals
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第18週 |
1/11 |
Final exam |
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