Math 145a Lecture Notes Evan Chen [email protected] Harvard College Fall 2014 This is Harvard College's Math 145a, instructed by Peter Koellner. The formal name for this class is \Set Theory I". The permanent URL is http://web.evanchen.cc/coursework.html, along with all my other course notes. Contents 1 September 2, 2014 5 1.1 Broad Overview . 5 1.2 Introduction to Independence . 5 1.3 Example of Independence . 5 1.4 More Surprising Examples . 6 1.5 Escaping . 7 1.6 Models and Compactness . 7 1.7 Looking Ahead . 7 1.8 So what happens in this class? . 8 2 September 4, 2014 9 2.1 Sets . 9 2.2 Subsets . 9 2.3 Cantor's Theorem . 9 2.4 Informal comprehension principle . 10 2.5 Empty set . 11 2.6 Well-Orderings . 11 2.7 Ordinals . 12 2.8 The Hierarchy . 12 3 September 9, 2014 13 3.1 Housekeeping . 13 3.2 Last Time . 13 3.3 Quasi-philosophical interlude . 13 3.4 Categoricity . 14 3.5 Language of Set Theory . 14 3.6 The First Five Axioms . 14 3.7 The Sixth Axiom, Infinity . 15 3.8 The Seventh Axiom, Foundation . 16 3.9 The Last Infinitely Many Axioms, Replacement . 16 1 Evan Chen [email protected] (Harvard College Contents Fall 2014) 4 September 11, 2014 17 4.1 Defining the nonnegative integers . 17 4.2 Comprehension and Collection . 17 4.3 Set abstraction and the \set of all sets" . 17 4.4 Some abuse of notation with sets and classes . 18 4.5 Some more basic notions . 19 4.5.1 Intersection and Difference . 19 4.5.2 Ordered Pairs . 19 4.5.3 Products . 19 4.5.4 Relations . 19 4.5.5 Functions . 20 4.6 Philosophy . 20 4.7 Axiom of Choice . 20 5 September 16, 2014 21 5.1 Motivation . 21 5.2 Setup . 21 5.3 Transfinite Induction . 22 5.4 Transfinite Recursion . 22 5.5 Transfinite things can be BIG . 23 6 September 18, 2014 25 6.1 What does a well-ordering look like? . 25 6.2 Well-orderings look roughly the same . 25 6.3 Stuff with classes . 26 6.4 How to preserve transitive sets . 27 6.5 Ordinals . 27 7 September 23, 2014 29 7.1 Recap . 29 7.2 Order Types . 29 7.3 Arithmetic on Ordinals . 30 7.3.1 Addition . 30 7.3.2 Multiplication . 30 7.3.3 Exponentiation . 31 7.3.4 A nontrivial theorem . 31 7.4 The hierarchy of sets, done with ordinals . 31 7.5 The ordinals grab all the sets . 32 8 September 25 33 8.1 Axiom of Choice is equivalent to WO . 33 8.2 Zorn's Lemma . 33 8.3 Some weakenings of AC . 34 9 September 30, 2014 35 9.1 Equinumerous sets . 35 9.2 Creating cardinals . 36 9.3 Properties of cardinals . 36 9.4 Recurse . 37 2 Evan Chen [email protected] (Harvard College Contents Fall 2014) 10 October 2, 2014 38 10.1 Cardinal arithmetic . 38 10.2 Finite sets . 38 10.3 Infinite cardinal arithmetic is boring . 39 10.4 Cardinal exponentiation . 39 10.5 Fundamental theorems of cardinal arithmetic . 40 10.6 Some exponentiation is not interesting . 40 11 October 7, 2014 41 11.1 Cofinality . 41 11.2 Lemmas on cofinality . 41 11.3 Given AC, cofinalities and successor cardinals are regular . 42 11.4 Inaccessible cardinals . 44 12 October 9, 2014 45 12.1 Linear Orders . 45 12.2 Intermission . 46 12.3 Separable complete linear orders . 46 12.4 Suslin property . 47 13 October 14, 2014 49 13.1 Sparse Review . 49 14 October 16, 2014 50 14.1 Partitions . 50 14.2 Infinite Ramsey Theorem . 50 14.3 Some negative results on partitions . 51 14.4 Large Cardinals . 52 14.5 Trees . 53 15 October 21, 2014 54 15.1 Definability . 54 15.2 Fragments of the Universe . 54 15.3 Definable sets . 55 16 October 23, 2014 57 16.1 Model theory notions . 57 16.2 Reflection . 58 16.3 Variations . 59 17 October 28, 2014 60 18 October 30, 2014 61 18.1 Completing the proof of a theorem . 61 18.2 The statement Lκ V = L .......................... 61 18.3 A Nice application of Reflection . 63 18.4 A sketch of the proof that Lκ AC ..................... 63 19 November 4, 2014 64 19.1 Axiom of Choice . 64 19.2 GCH . 64 19.3 Hulls . 65 3 Evan Chen [email protected] (Harvard College Contents Fall 2014) 19.4 Summary . 66 20 November 6, 2014 67 20.1 β-Gaps . 67 20.2 A Taste of Fine Structure . 67 20.3 The Guessing Sequence . 68 21 November 11, 2014 70 21.1 Problems with trying to set up Forcing . 70 21.2 Coding; good and bad reals . ..