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Set Theory and Logic at Part II in 2016/7 Set Theory and Logic at Part II in 2016/7 Thomas Forster May 17, 2017 Contents 1 Lecture 1: First of Five Lectures on Ordinals and Cardinals 5 1.1 Ordinals . .5 1.2 Wellfoundedness . .6 2 Lecture 2 9 3 Lecture 3 11 4 Lecture 4 13 5 Lecture 5: First of 5 lectures on Posets 14 6 Lecture 6: Fixed-point theorems 16 6.1 The Axiom of Choice . 17 7 Lecture 7 18 7.0.1 Weak versions: countable choice, and a classic application thereof; DC . 19 7.0.2 Applications of Zorn's Lemma . 19 8 Lecture 8: Boolean Algebras Continued 19 8.1 Reduced products . 21 9 Lecture 9: First of Five Lectures on Propositional Logic 21 9.1 if-then . 22 10 Lecture 10 26 11 Lecture 11 29 11.1 Boolean algebras detect propositional tautologies . 30 12 Lecture 12 32 12.1 Applications of Propositional Compactness . 32 12.1.1 The Interpolation Lemma . 33 1 13 Lecture 13 34 13.1 CNF and DNF . 34 13.1.1 Resolution Theorem Proving . 35 13.2 Predicate Logic begun . 36 13.3 The Syntax of First-order Logic . 36 13.3.1 Constants and variables ................. 36 13.3.2 Predicate letters ...................... 36 13.3.3 Quantifiers ......................... 37 13.3.4 Function letters ...................... 37 14 Lecture 14 38 14.1 Axioms . 39 14.2 Semantics . 40 14.3 Completeness theorem for LPC: the set of valid sentences is semide- cidable . 43 14.3.1 2-terms . 43 15 Lecture 15 43 15.1 Decidability . 46 16 Lecture 16 46 16.1 The Skolem-L¨owenheim Theorems . 46 16.2 Categoricity . 47 16.2.1 Failure of Completeness of Second-order Logic . 47 16.3 Results related to completeness, exploiting completeness . 49 16.3.1 Prenex Normal Form and quantifier-counting . 49 17 Lecture 17: The Same Continued 50 17.1 Omitting Types . 51 17.2 Some Set Theory . 51 18 Lecture 18 53 18.1 Transitive Closures and Transitive Sets . 54 18.2 The Cumulative Hierarchy . 55 18.3 Scott's Trick . 56 19 Lecture 19 57 19.1 The Axiom of Foundation . 57 19.2 Mostowski Collapse . 58 19.3 Ordinals again . 60 19.3.1 Initial ordinals . 61 20 Lecture 20 62 20.1 @2 = @ ................................. 63 2 21 Lecture 21 65 21.1 Independence of the Axioms from each other . 66 21.1.1 ∆0 formulae and the L´evyHierarchy . 67 21.1.2 Some actual independence results . 68 22 Lecture 22 (Independence results continued) 70 22.0.3 Independence of Sumset . 72 22.1 Independence of the Axiom of Foundation . 72 22.2 Independence of the Axiom of Choice . 73 23 Lecture 23 76 24 Lecture 24: Independence 76 25 Example Sheets 77 The Toad never answered a word, or budged from his seat in the road; so they went to see what was the matter with him. They found him in a sort of trance, a happy smile on his face, his eyes still fixed on the dusty wake of their destroyer. At intervals he was still heard to murmer `Poop-poop!' Kenneth Graham The Wind in The Willows, chapter 2. 3 The Rubric LOGIC AND SET THEORY (D) 24 lectures, Lent term No specific prerequisites. Introduction Ordinals and cardinals Well-orderings and order-types. Examples of countable ordinals. Uncount- able ordinals and Hartogs' lemma. Induction and recursion for ordinals. Ordinal arithmetic. Cardinals; the hierarchy of alephs. Cardinal arithmetic. [5] Posets and Zorn's lemma. Partially ordered sets; Hasse diagrams, chains, maximal elements. Lattices and Boolean algebras. Complete and chain-complete posets; fixed-point theo- rems. The axiom of choice and Zorn's lemma. Applications of Zorn's lemma in mathematics. The well-ordering principle. [5] Propositional logic The propositional calculus. Semantic and syntactic entailment. The deduc- tion and completeness theorems. Applications: compactness and decidability.[3] Predicate logic The predicate calculus with equality. Examples of first-order languages and theories. Statement of the completeness theorem; *sketch of proof*. The com- pactness theorem and the L¨owenheim-Skolem theorems. Limitations of first- order logic. Model Theory. [5] Set theory Consistency Problems of consistency and independence. [1] Appropriate books J.L Bell and A Slomson. Models and Ultraproducts North Holland 1969 recently reissued by Dover B.A. Davey and H.A. Priestley Lattices and Order. Cambridge University Press 2002 T. Forster Logic, Induction and Sets. Cambridge University Press A. H´ajnaland P. Hamburger Set Theory. LMS Student Texts number 48, CUP 1999 A.G. Hamilton Logic for Mathematicians. Cambridge University Press 1988 P.T. Johnstone Notes on Logic and Set Theory. Cambridge University Press 1987 D. van Dalen Logic and Structure. Springer-Verlag 1994 4 1 Lecture 1: First of Five Lectures on Ordinals and Cardinals [Lecture 1: First of Five Lectures on Ordinals and Cardinals, preceded by some introductory patter] (The rubric says): Well-orderings and order-types. Examples of countable ordinals. Uncountable ordinals and Hartogs' lemma. In- duction and recursion for ordinals. Ordinal arithmetic. Cardinals; the hierarchy of alephs. Cardinal arithmetic. (Some of the arithmetic of ordinals and cardinals cannot really be done properly until we have some set theory under our belt. OTOH it's good to at least introduce the students to these ideas early on in the piece. The resulting exposition is inevitably slightly disjointed.) Warning! If you are reading this and your name isn't `Thomas Forster' then you are eavesdropping; these notes are my messages to myself and are made available to you only on the off-chance that such availability might help you in preparing your own notes for this course. This warning doesn't mean that you shouldn't be reading this document, but you should bear it in mind anyway because i do not write out here in detail things i can do off the top of my head. I write out things than i might, in the heat of the moment, get wrong, or do in the wrong order|or forget altogether. However it is my settled intention that by fairly early in the Lent Term these notes will be in a state fit for students to use them to revise from.. 1.1 Ordinals Cantor's discovery of a new kind of number. 1IR 6= 1IN etc etc. 1IR is a multi- plicative unit whereas 1IN is the quantum of multiplicity (\how many?"). Brief chat about datatypes. \How many times do i have to tell you to tidy up your room?" the answer will be an ordinal (possibly finite). Cantor's discussion of closed sets of reals. Ordinals measure the length of discrete deterministic monotone pro- cesses. (synchronous/asynchronous doesn't matter) Well, we mean something slightly more than discrete . the set of stages has a total order, and it's always the case that the set of unreached stages has a first element. Monotonicity ensures that it's always clear what the situation is that you are in, and determinism-and-discreteness means that there is always an immediately-next thing to do and that you know what it is. !, ! + n, ! + !, ! · n, ! · !!n. Ordinals are also the order types of special kinds of total orders. ! is the order-type of hIN; <INi. (I write structures as tuples, carrier set followed by 5 operations). Pick 0 off the front and put it on the end, get a bigger ordinal| but the underlying set is the same size. In fact we get ! + 1, which illustrates how addition corresponds to concatenation. But to understand order types we need to put the project into a more general context: We need to do some logic. DEFINITION 1 Congruence relations. Congruence relations give rise to operations on the quotient. Cardinals are very simple. Read my countability notes. Multiplication, addition and exponentiation. 1.2 Wellfoundedness Suppose we have a carrier set with a binary relation R on it, and we want to be able to infer 8x (x) from (8x)((8y)(R(y; x) ! (y)) ! (x)) In words, we want to be able to infer that everything is from the news that you are as long as all your R-predecessors are . y is an R-predecessor of x if R(y; x). Notice that there is no \case n = 0" clause in this more general form of induction: the premiss we are going to use implies immediately that a thing with no R-predecessors must have . The expression \(8y)(R(y; x) ! (y))" is called the induction hypothesis. The first line says that if the induction hypothesis is satisfied, then x is too. Finally, the inference we are trying to draw is this: if x has whenever the induction hypothesis is satisfied, then everything has . When can we do this? We must try to identify some condition on R that is equivalent to the assertion that this is a legitimate inference to draw in general (i.e., for any predicate ). Why should anyone want to draw such an inference? The antecedent says \x is as long as all the immediate R-predecessors of x are ", and there are plenty of situations where we wish to be able to argue in this way. Take R(x; y) to be \x is a parent of y", and then the inference from \children of blue-eyed parents have blue eyes" to \everyone has blue eyes" is an instance of the rule schematised above. As it happens, this is a case where the relation R in question does not satisfy the necessary condition, for it is in fact the case that children of blue-eyed parents have blue eyes and yet not everyone is blue-eyed.
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