Last modified by asaekman@helsinki_fi on 2024/03/27 10:47
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1 = Logic toolbox for mainstream mathematicians, fall 2015 =
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3 {{note}}
4 Lectures and exercise class on Thursday 15.10. and Friday 1610. cancelled (lecturer still ill). Also the exam is postponed - see your mail for details.
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11 {{panel}}
12 **Teacher:** [[Åsa Hirvonen>>doc:mathstatHenkilokunta.Hirvonen, Åsa]]
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14 **Scope:** 5 cr
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16 **Type:** Advanced studies
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18 **Teaching: **Period I, lectures Mon 10-12, Thu 14-16, exercise class Fri 10-12.
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20 (% align="LEFT" class="western" %)
21 **Topics:** We look at methods from logic useful in other areas of mathematics, such as basic cardinal arithmetic, transfinite induction, ultraproducts and, time permitting, Martins Axiom.
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23 **Prerequisites:** The course does not require previous knowledge of logic, but some 'mathematical routine' is assumed. (the course is primarily aimed at master's and graduate students)
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26 === {{toc maxLevel="4" minLevel="2" indent="20px"/}} ===
27
28 == News ==
29
30 * The first lecture is on Monday 31.8. at 10-12
31 * We start exercises already the first week with warm up exercises (done on site).
32
33 == Teaching schedule ==
34
35 Weeks 36-42, Monday 10-12 and Thursday 14-16 in room B120. In addition, two hours of exercise classes per week.
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37 == Exams & project ==
38
39 There will be a final exam of the course on **Wednesday 21.10. at 12-14.30 in the large auditorium (A111).**
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41 **The deadline for the project is 1.11.2015.**
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43 The course is evaluated based on the exam (max. 24p), a project work (max 12p) and the exercises (max 6p). The project can be rather freely chosen as long as it relates to the theme of using logical tools in mathematics. It should be around 3-4 typed pages long. Examples of suitable projects are
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45 * the Cantor-Bendixon rank
46 * some not too trivial application of transfinite induction (e.g. Goodstein's theorem)
47 * Fodor's lemma
48 * comparing Banach space ultraproducts to ultraproducts
49 * infinitesimals via ultraproducts
50 * applications of MA in analysis (or some other suitable field)
51
52 == Course material ==
53
54 [[Ordinals>>attach:Ordinals.pdf]] (corrected 3.9.)
55 [[Transfinite induction and recursion
56 >>attach:Transfinite_induction.pdf]][[Cardinals
57 >>attach:Cardinals.pdf]][[Ultraproducts>>attach:Ultraproducts.pdf]]
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59 [[Appendix: ZFC axioms>>attach:Axioms.pdf]]
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61 Lecture notes will appear here during the course. For a more thorough treatment (or a sneak preview of the subjects) you can consult e.g.
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63 H. Enderton: Elements of set theory, Academic press. (intro to set theory; thorough intro to ordinals and cardinals)
64 K. Kunen: Set Theory An Introduction to Independence Proofs, Elsevier. (more set theory; cardinal arithmetic and Martin's axiom)
65 C. C. Chang, H. J. Keisler: Model Theory, Elsevier. (model theory; ultraproducts, also has an intro to ordinals and cardinals in the appendix)
66
67 == [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57072||shape="rect"]] ==
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70 (% style="color: rgb(96,96,96);" %)Did you forget to register? (%%)[[What to do?>>url:https://wiki.helsinki.fi/display/mathstatOpiskelu/Kysymys4||style="text-decoration: underline;" shape="rect"]]
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72 == Exercises ==
73
74 === Assignments ===
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76 * [[Exercises 1>>attach:h1.pdf]] (warm-up exercises, done on site)
77 * [[Exercises 2>>attach:h2.pdf]] (corrected 8.9.)
78 * [[Exercises 3>>attach:h3.pdf]]
79 * [[Exercises 4>>attach:h4.pdf]]
80 * [[Exercises 5>>attach:h5.pdf]]
81 * [[Exercises 6>>attach:h6.pdf]]
82 * [[Exercises 7>>attach:h7.pdf]]
83
84 === Exercise classes ===
85
86 |=(((
87 Group
88 )))|=(((
89 Day
90 )))|=(((
91 Time
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93 Room
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95 Instructor
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97 |(((
98 1.
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100 Friday
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102 10-12
103 )))|(((
104 C123
105 )))|(% colspan="1" %)(((
106 Åsa Hirvonen
107 )))
108
109 == Logbook ==
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111 31.8. Ordinals.
112 3.9. Transfinite induction
113 7.9. Transfinite recursion, cardinality
114 10.9. Cardinal arithmetic
115 14.9. more on cardinal arithmetic; use in induction
116 17.9. cofinality
117 21.9. stuctures and filters
118 24.9. ultraproducts
119 28.9. Los's theorem
120 1.10. Applications of Los's theorem (compactness, non-axiomatizability)
121 5.10. Martin's Axiom
122 \\
123
124 == Course feedback ==
125
126 Course feedback can be given at any point during the course. Click [[here>>url:https://elomake.helsinki.fi/lomakkeet/11954/lomake.html||style="line-height: 1.4285;" shape="rect"]].