課程名稱 |
基本模型理論 Elementary Model Theory |
開課學期 |
109-2 |
授課對象 |
文學院 哲學研究所 |
授課教師 |
楊金穆 |
課號 |
Phl7712 |
課程識別碼 |
124EM2980 |
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。研究所:C領域。
大學部:(C)哲學專題群組。 總人數上限:15人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1092Phl7712_ |
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 |
作為數理邏輯的重要分支,模型理論簡而言之旨在處理形式語言和模型之間的關係。此處的「形式語言」指任何適用於一階邏輯理論或數學理論的理論,而所謂的「模型」則指能用以詮釋語言,滿足以形式語言表達的邏輯真理、數學真理或種種關係的抽象結構。換言之,從數學的角度來看,模型理論研究如何為特定的數學理論(如實分析、代數等等)建構其適用的模型,並比較模型之間的關係與特點。因此,模型理論可視為一門邏輯與數學的交岔學科,企圖刻劃數學結構的一般邏輯形式及特徵。
儘管如此,由於本課程屬於導論性質的課程,而主要授課對象並非數學專業的同學,因此課程內容會略過關於特定數學結構(如群、環、代數等)的討論,著重於語言與模型的關係,以及幾種重要且能引發哲學討論的建模方式。一開始,會先重溫一下基本邏輯和中階邏輯的內容,介紹如何設計適用的形式語言(特別是一階形式語言),並引進著名塔斯基語意學(Tarski’s formal semantics),據此定義邏輯的真概念之後,再利用亨金(Henkin)的方法,證明一階邏輯系統的完備性定理(completeness theorem);接著會談滿足一階形式語言的模型具有哪些特點,並介紹幾種不同的建模方法及其涉及的重要邏輯工具和技巧;本課程的最後會稍微介紹一下這些邏輯工具和技巧如何應用於賽局理論。
本課程的授課對象為大二以上到研究所程度,對邏輯、數學、語言哲學、數學哲學、科學哲學等領域感興趣的學生。已經學過基本邏輯、中階邏輯、基本集合論或相關概念(如形式語言、述詞邏輯、在某模型中為真、可靠性定理、完備性定理、基數、序數、選擇公理、佐恩引理、良序原則、可數集、不可數集、同構、連續統假設)的學生修習本課程,會比較容易進入狀況。不過,本課程會視學生能力,適時補充相關基礎知識。 |
課程目標 |
對語言和模型的關係有初步的認識,熟悉著名的模型理論定理,並能應用本課程內容從事邏輯哲學、數學哲學、語言哲學或科學哲學方面的研究。 |
課程要求 |
課前閱讀講義,上課提問,並參加期末考試。 |
預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
楊金穆編,《基本模型理論》。
Chin-Mu Yang, ed. (2015) The Basic Model Theory. (Lecture Notes) |
參考書目 |
Badesa, Calixto (2004), The Birth of Model Theory─Lowenheim’s Theorem in the Frame of the Theory of Relatives, Princeton, N. J.: Princeton University Press.
Bridge, J. (1977), Beginning Model Theory, Oxford: Clarendon Press.
Chang, C. C. and Keisler, H. J. (1990), Model Theory (3rd ed., especially Chapaters 1-3 and 4.1), Amsterdam: North-Holland.
Chatzidakis, Z. et al. (eds.), 2008, Model Theory with Applications to Algebra and Analysis, Volumes 1 and 2, Cambridge: Cambridge University Press.
Doets, K. (1996), Basic Model Theory, Stanford, California: CSLI.
Fra?sse, R. (1974), Course of Mathematical Logic Vol 2 Model Theory, Dordrecht-Holland: D. Reidel.
Hart, B., Lachlan, A. and Valeriote, M., 1996, Algebraic Model Theory, Dordrecht: Kluwer.
Haskell, D., Pillay, A. and Steinhorn, C., 2000, Model Theory, Algebra, and Geometry, Cambridge: Cambridge University Press.
Hodges, W. (1993), Model Theory (Encyclopedia of mathematics and its applications, Vol. 42), Cambridge: Cambridge University Press.
Hodges, W. (1997), Shorter Model Theory, Cambridge: Cambridge University Press.
Keisler, H. J. (1977), ‘Fundamentals of model theory’, Chapter A.2 of Handbook of Mathematical Logic, J. Barwise (ed.), Amsterdam: North Holland.
Manzano, Maria (1999), Model Theory (Oxford Logic Guides Vol. 37- Oxford Science publications), Oxford: Oxford University Press.
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評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
Weekly answers/Short essay |
30% |
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2. |
Long essay |
40% |
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3. |
Performance in class-discussion |
30% |
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週次 |
日期 |
單元主題 |
第1週 |
2/26 |
Introduction
Model theory: Models of theories and structures
Relational structures (Abstract models)
Theories and models: Proof-theoretical vs. model-theoretical constructions of a theory
The development of model theory: A historical review
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第2週 |
3/05 |
The predicate calculus— A proof-theoretical approach to predicate logic
Elements of first-order language
A naïve e first-order language LQ suitable for predicate logic
The predicate calculus— A proof-theoretical construction
Basic semantics for predicate logic
Substitutional interpretation of quantification
Objectual interpretation of quantification and its structures
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第3週 |
3/12 |
The predicate calculus - A model-theoretical approach to predicate logic
Types of relational structures and similarity types
A similarity type σ of structures suitable for predicate logic, and its language Lσ
Semantics for the language Lσ —A Tarskian style semantics for first-order languages
The predicate calculus
A Henkin-style proof of the completeness theorem for the predicate calculus
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第4週 |
3/19 |
Basic Maps between structures, and some semantic consequences
Substructures, extensions, embeddings, homomorphism and isomorphism
Elementary equivalence, elementary substructures and elementary embeddings
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第5週 |
3/26 |
Compactness theorem and Löwenheim-Skolem theorems
The compactness theorem for first-order language
The Löwenheim-Skolem theorems
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第6週 |
4/02 |
Theories of structures and unions of chains
Model-theoretical construction of theories
Chains of structures: elementary extensions and elementary chains
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第7週 |
4/09 |
Diagrams |
第8週 |
4/16 |
Some results of unions of chains and the method of diagrams |
第9週 |
4/23 |
midterm break |
第10週 |
4/30 |
Categoricity
Complete theories and categoricity
Back and forth property
o-categoricity
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第11週 |
5/07 |
Types and omitting type theorems;
Types
Omitting types
Ryll-Nardzewski theorem
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第12週 |
5/14 |
Existentially closed structures: Inductive models and model-theoretic forcing
Existentially closed structures and model-completeness
Inductive structures
Model-theoretical forcing
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第13週 |
5/21 |
Ultraproducts and Henkin-Keisler models
Filters and ultrafilters
Reduced products and ultraproducts
Henkin-Keisler models
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第14週 |
5/28 |
Countable models of complete theories: atomic models and saturated models
Atomic models and prime models
Saturated models
Recursively saturated models and homogenous models
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第15週 |
6/04 |
Game theory: An introduction |
第16週 |
6/11 |
Finite two person games |
第17週 |
6/18 |
Game-theoretic semantics |
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