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Lectures on Infinitary Model Theory [Kõva köide]

(University of Illinois, Chicago)
  • Formaat: Hardback, 192 pages, kõrgus x laius x paksus: 236x160x18 mm, kaal: 430 g, Worked examples or Exercises
  • Sari: Lecture Notes in Logic
  • Ilmumisaeg: 27-Oct-2016
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107181933
  • ISBN-13: 9781107181939
Teised raamatud teemal:
Lectures on Infinitary Model TheorySuurem pilt
  • Formaat: Hardback, 192 pages, kõrgus x laius x paksus: 236x160x18 mm, kaal: 430 g, Worked examples or Exercises
  • Sari: Lecture Notes in Logic
  • Ilmumisaeg: 27-Oct-2016
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107181933
  • ISBN-13: 9781107181939
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107181939.html
Märksõnad:
Infinitary logic, the logic of languages with infinitely long conjunctions, plays an important role in model theory, recursion theory and descriptive set theory. This book is the first modern introduction to the subject in forty years, and will bring students and researchers in all areas of mathematical logic up to the threshold of modern research. The classical topics of back-and-forth systems, model existence techniques, indiscernibles and end extensions are covered before more modern topics are surveyed. Zilber's categoricity theorem for quasiminimal excellent classes is proved and an application is given to covers of multiplicative groups. Infinitary methods are also used to study uncountable models of counterexamples to Vaught's conjecture, and effective aspects of infinitary model theory are reviewed, including an introduction to Montalbán's recent work on spectra of Vaught counterexamples. Self-contained introductions to effective descriptive set theory and hyperarithmetic theory are provided, as is an appendix on admissible model theory.

Muu info

Explores connections between infinitary model theory and other branches of mathematical logic, with algebraic applications.
Introduction 1(6)
Part 1 Classical results in infinitary model theory
Chapter 1 Infinitary languages
7(8)
1.1 Fragments and Downward Lowenheim--Skolem
10(3)
1.2 Lω1.ω and omitting first order types
13(2)
Chapter 2 Back and forth
15(14)
2.1 Karp's Theorem
15(3)
2.2 Scott's Theorem
18(5)
2.3 Countable approximations
23(3)
2.4 Larger infinitary languages
26(3)
Chapter 3 The space of countable models
29(10)
3.1 Spaces of τ-structures
29(2)
3.2 The number of countable models
31(3)
3.3 Scattered sentences and Morley's proof
34(5)
Chapter 4 The Model Existence Theorem
39(12)
4.1 Consistency properties and model existence
39(3)
4.2 Omitting types and atomic models
42(4)
4.3 The Interpolation Theorem
46(3)
4.4 The undefinability of well-ordering
49(2)
Chapter 5 Hanf numbers and indiscernibles
51(14)
5.1 The Erdos--Rado Partition Theorem
52(1)
5.2 The Hanf number of Lω1.ω
53(3)
5.3 Morley's Two Cardinal Theorem
56(1)
5.4 Completely characterizing ℵ1
57(8)
Part 2 Building uncountable models
Chapter 6 Elementary chains
65(18)
6.1 Elementary end extensions
65(5)
6.2 Omitting types in end extensions
70(3)
6.3 Uncountable models realizing few types
73(4)
6.4 Extending models of set theory
77(2)
6.5 ℵ1-categorical sentences have models in ℵ2
79(4)
Chapter 7 Vaught counterexamples
83(14)
7.1 Minimal counterexamples
83(4)
7.2 Harrington's Theorem
87(10)
Chapter 8 Quasiminimal excellence
97(18)
8.1 Categoricity
97(8)
8.2 Covers of Cx
105(10)
Part 3 Effective considerations
Chapter 9 Effective descriptive set theory
115(12)
9.1 Recursion theory review
115(2)
9.2 Computable functions on ωω
117(1)
9.3 The arithmetic hierarchy
118(4)
9.4 The effective projective hierarchy
122(1)
9.5 Recursive ordinals
123(4)
Chapter 10 Hyperarithmetic sets
127(12)
10.1 Borel codes
127(1)
10.2 Recursively coded Borel sets
128(3)
10.3 Hyperarithmetic sets
131(1)
10.4 The Effective Perfect Set Theorem
132(7)
Chapter 11 Effective aspects of Lω1.ω
139(12)
11.1 Coding Lω1.ω-formulas
139(2)
11.2 Kreisel--Barwise Compactness
141(4)
11.3 Effective analysis of Scott rank
145(6)
Chapter 12 Spectra of vaught counterexamples
151(8)
12.1 Determinacy and Turing degrees
151(4)
12.2 Montalban's Theorem
155(4)
Appendix A ℵ1-free abelian groups
159(4)
Appendix B Admissiblility
163(12)
B.1 Kripke--Platek set theory
163(3)
B.2 Admissible sets and HYP
166(1)
B.3 Model theory of admissible fragments
167(6)
B.4 C1 = Q
173(2)
References 175(6)
Index 181
David Marker is LAS Distinguished Professor of Mathematics at the University of Illinois, Chicago, and a Fellow of the American Mathematical Society. His main area of research is model theory and its connections to algebra, geometry and descriptive set theory. His book, Model Theory: An Introduction, is one of the most frequently used graduate texts in the subject and was awarded the Shoenfield Prize for expository writing by the Association for Symbolic Logic.