Notes for MATH 582 David J. Fern´andez-Bret´on (based on the handwritten notes from January-April 2016, typed in January-April 2017, polished in January-April 2018) Table of Contents Chapter 1 Basic set theory 3 1.1 What is a Set? . 3 1.2 Axiomatizing set theory . 3 1.3 The Language of set theory . 4 1.4 Cantor's set theory . 6 1.5 Patching up Cantor's set theory . 8 1.6 Other ways of dodging Russell's Paradox . 14 1.7 The last few words about the axiomatic method . 15 Chapter 2 Relations, functions, and objects of that ilk 17 2.1 Ordered pairs, relations and functions . 17 2.2 Equivalence relations . 23 2.3 Partial orders . 26 2.4 Well-orders . 28 Chapter 3 The natural numbers, the integers, the rationals, the reals... and more 32 3.1 Peano Systems . 32 3.1.1 Arithmetic in a Peano system . 34 3.1.2 The order relation in a Peano system . 35 3.1.3 There exists one Peano system . 37 3.2 The integers . 38 3.3 The rational numbers . 38 3.4 The real numbers . 39 3.4.1 Dedekind cuts . 39 3.4.2 Cauchy sequences . 39 3.5 Other mathematical objects . 44 Chapter 4 Ordinal numbers 45 4.1 Definition and basic facts . 45 4.2 Classes . 48 4.3 A few words about NBG ............................................... 50 4.4 The Axiom of Replacement . 50 4.5 Ordinal numbers as representatives of well-ordered sets . 53 4.6 Sets and their ranks . 53 4.7 Ordinal arithmetic . 54 Chapter 5 Cardinality 55 5.1 Equipotence, countable and uncountable sets . 55 5.2 A partial order among cardinalities . 57 5.3 The continuum, the continuum hypothesis, the @ sequence . 58 5.4 Cardinal Arithmetic . 60 Chapter 6 The Axiom of Choice 61 6.1 Equivalences of the Axiom of Choice . 61 6.2 Cardinal numbers after the Axiom of Choice (two more equivalences) . 67 6.3 The Banach-Tarski paradox . 69 6.4 Dedekind-finite sets . 71 6.5 Some more cardinal arithmetic . 73 6.6 The canonical well-ordering of Ord × Ord and its consequences . 73 1 6.7 The last few equivalences . 74 Chapter 7 Well-founded relations and the Axiom of Foundation 76 7.1 Well-founded relations . 76 7.2 The Axiom of Foundation . 78 Appendix A Axioms of Set Theory 80 Appendix B Worksheet 1: Constructing Z 81 Appendix C Worksheet 2: Constructing Q 83 Appendix D Worksheet 3: Ordinal arithmetic 85 Appendix E Worksheet 4: Na¨ıve cardinal arithmetic 86 Appendix F Worksheet 5: The Axiom of Choice 88 Appendix G Statements equivalent to the Axiom of Choice 90 2 Chapter 1 Basic set theory Set theory is sometimes considered to be a branch of Mathematical Logic. In particular, it might be relevant not just for mathematics, but for computer science and philosophy as well. It is often said that set theory was born \on that December of 1873 in which Cantor proved that there are uncountably many reals". It is certainly true that Cantor must have been one of the first few people which considered abstract collections of mathematical objects at the level of abstraction that we now do in set theory. Over the course of the years, set theory has blossomed and developed to the point where it now constitutes a whole branch of mathematics on its own. This course is an introduction to this exciting area of mathematics. This course has two main objectives. The first is to convince students that every mathematical object (including: functions, (real and complex) numbers, points in n-dimensional space, polynomials, and virtually everything that you can think of) can be viewed as a set of some sort (some people like to say that \everything is a set", although I consider it to be more accurate if we say that everything can be implemented within set theory). Thus we will spend the first half of this course introducing the commonly accepted axioms of set theory, and explaining how, starting from these axioms, it is possible to implement most everyday mathematical objects by defining appropriate sets that behave, when interpreted in the appropriate way, as we would intuitively expect these objects to behave. The culmination of this will be the construction of the Real Line R. The second objective of this course is to introduce the student to a few basic topics of set theory proper, most notably Ordinal Numbers (and Transfinite Induction and Recursion), Infinite Cardinal Arithmetic (with some level of sophistication, beyond the elementary and simple-minded distinction between countable and uncountable) and some equivalences and consequences of the Axiom of Choice (including applications to other areas of Mathematics). These topics will roughly constitute the second half of the course. Recently, I have started referring to the topics from the first half as the \foundational" part of set theory, whereas the topics from the second half constitute a fragment of the \mathematical" part of set theory. 1.1 What is a Set? Informally, we can think of a set as some collection of objects, called its elements, where the word \collection" is intended in a very abstract way. A set is usually written either explicitly as the list of its elements (e.g. the set f6; 28g), or as a description of what its elements look like (e.g. the set fn n is a perfect number and n ≤ 100g). The special symbol 2 is introduced as a binary relation, stating that the object to the left of the symbol is an element of the set to the right of it: for example, 6 2 f6; 28g. Formally speaking, though, it is not necessary to explicitly write down any definition of what a set is: a set would just be any object belonging to set theory, and all that is therefore relevant is that these objects behave like the axioms of set theory say that they do. In other words, the axioms themselves play, to a certain extent, the r^oleof definitions (in the same sense that, for example, the group axioms constitute the very definition of what a group is, rather than being some \obvious" statements about some predetermined mathematical object). So the really important part, to begin with our study of sets, is to explicitly state what are the axioms of set theory. 1.2 Axiomatizing set theory There are a number of possible axiomatizations of set theory that have been proposed over the course of the years. The main ones among these are: • The axioms of Zermelo-Fraenkel, along with the Axiom of Choice (abbreviated ZFC), • the axioms of von Neumann-Bernays-G¨odel(abbreviated NBG), 3 • the axioms of Morse-Kelley (abbreviated MK), • Quine's New Foundations (abbreviated NF), • the modification of NF that allows objects known as urelements (abbreviated NFU). Mostly for historical reasons, the axiom system that is currently the most widespread and commonly accepted is ZFC, and so these are the axioms that we will use in this course, although I will briefly mention the other axiom systems, and how they differ from ZFC, in appropriate moments throughout the course. So in theory, we should now proceed to state what the ZFC axioms are. However, the list of ZFC axioms consists of seven axioms and two axiom schemas (resulting in infinitely many axioms overall), which might look quite arbitrary at first sight. So in order to motivate these axioms, we will first spend some time working in what I like to call \Cantor's set theory", see what are its advantages and also its disadvantages, and what problems arise from this way of doing set theory. After this, the ZFC axioms should (for the most part) look quite natural. What is Cantor's set theory, then? Cantor used to do set theory based on only two very basic principles. The first one is the following: Axiom 1 (Principle of Extensionality). A set is determined by its elements. This means that, if A and B are two sets that have the exact same elements, then they are actually the same set, symbolised A = B. This principle is central to the concept of a set, as it explicitly expresses that the identity of a set is given only by the objects that are elements of this set, irrespective of any other property (such as the order in which its elements are written, or thought of; or the number of times that we write such elements). For example, it is based on this principle that we can conclude that the following four sets are actually the same: f6; 28g = f28; 6g = f6; 28; 6g = fn n is a perfect number and n ≤ 100g: The second basic principle in Cantor's set theory is the following: Axiom 2 (Principle of Set Creation). Given any property, there exists a set whose elements are exactly those objects that satisfy the given property. Thus, if we denote a property of some object x by P (x), then the Principle of Set Creation ensures the existence of a set A such that x 2 A if and only if P (x). As we said before, this set is denoted by fx P (x)g, and the Principle of Extensionality ensures that it is unique. For example, if the property P (x) is \x is a prime number and x ≤ 30", then the Principle of Set Creation ensures the existence of the set fx P (x)g = f2; 3; 5; 7; 11; 13; 17; 19; 23; 29g: 1.3 The Language of set theory Now, there are two problems with Cantor's set theory, which we will mention in reverse (according to when they were historically pointed out) order.