A SET-THEORETIC APPROACH TO OBTAINING INFINITY by MATTHEW MICHAEL JONES B.A., University of Colorado, Colorado Springs, 2013 A thesis submitted to the Graduate Faculty of the University of Colorado Colorado Springs in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics 2016 ii This thesis for the Master of Science degree by Matthew Michael Jones has been approved for the Department of Mathematics by Greg Oman Gene Abrams Greg Morrow Date: 12/14/2016 iii Jones, Matthew Michael (M.S., Mathematics) A Set-Theoretic Approach To Obtaining Infinity Thesis directed by Assistant Professor of Mathematics, Dr. Greg Oman ABSTRACT Many set-theoretic axioms have been formulated over the years which imply (along with the other axioms for set theory) the existence of what one would intuitively regard as an infinite set. In this thesis, we identify a property common to many such axioms, and then we obtain a more general class of “infinity axioms.” iv TABLE OF CONTENTS CHAPTER I. INTRODUCTION 1 II. PRELIMINARIES 6 III. RESULTS 16 REFERENCES 25 1 CHAPTER I INTRODUCTION The concept of “infinity” is ubiquitous in mathematics. Indeed, the collection of real numbers, the collection of prime numbers, and the collection of real-valued, continuous functions are all examples of infinite collections of objects. This is a notion we intuitively understand: a collection of objects is infinite if, in some sense, its elements “go on forever.” In modern mathematics, this is hardly precise enough to be a formal definition. But how can one define infinity? There are many answers in the literature. The most common context in which such answers appear is in the mathematical field of set theory. Informally, a set is a collection of objects. For instance, {1, 2, 3} denotes the collection (set) of numbers 1, 2, and 3. It is generally recognized that the theory of sets is robust enough to encapsulate all of mathematics. Hence it is natural to investigate notions of infinity within this theory. A brief history follows. Mathematician Richard Dedekind produced many classical writings between the years 1871 and 1888 which promoted a set-theoretic formulation of mathematical ideas. In 1888 he published a presentation of the basic elements of (a somewhat primitive) set theory, making a bit more explicit the operations on sets and mappings he had been using since 1871. His definition of an infinite set is given below. Definition 1. (Dedekind Infinite Set) Let X be a set. Then X is Dedekind Infinite if and only if there exists a bijection between A and X for some proper subset A of X. 2 One may argue that Dedekind’s definition of an infinite set conforms to our informal notions of infinity. Indeed, let us consider some intuitively “infinite” set X. We will informally argue there is a bijection between A and X for some proper subset A of X. Indeed, since X is in some sense infinite, we can pick x1,x2,x3,... ∈ X. We now define f : X → X \{x1} by x if x∈ / {x1,x2,x3,...}, and f(x)= . xi+1 if x = xi. It is easy to see that f is a bijection between X and X\{x1}. Thus X is Dedekind Infinite. Conversely, one can give an argument (which is necessarily informal at this point since we have yet to introduce the ZFC axioms) that if X is not infinite, then X is not Dedekind Infinite. Another great mathematician, Paul St¨ackel, also made contributions to set theory. While at the University of Hannover in 1907, St¨ackel gave us his definition of an infinite set. Definition 2. (St¨ackelInfinite Set) Let X be a set. Then X is St¨ackelInfinite if and only if there exists a well-order on X such that the reversal order is not a well-order on X. In the next chapter we discuss well-orders in more detail and prove the Well- Ordering Theorem which states that every set can be well-ordered. Assuming (for now) the Well-Ordering Theorem, we note how St¨ackel’s definition also seems to capture our informal notion of what an infinite set should be. Indeed, let us consider some set X which is, informally, “infinite.” By the Well-Ordering Theorem, there exists a well-order < on X. Thus there must exist a least element of X, call it x1. 3 There must also exist a least element of X \{x1}, call it x2. We can continue this construction indefinitely as X is “infinite” to create the set S := {x1,x2,...}. It is clear that S has no greatest element, and thus in the reversal order, there is no least element. Hence X is a St¨ackel Infinite Set. Another contributor to the foundations of set theory was Alfred Tarski. He was a brilliant mathematician; in fact, he was the youngest person ever to complete a doctorate at Warsaw University. Tarski’s mathematical interests were exceptionally broad, yet his first paper, published in 1920 when he was 19 years old, was on set theory. In what follows, we let P(X) denote the power set of X, that is, the set of all subsets of X. We now give Tarski’s definition of an infinite set. Definition 3. (Tarski Infinite Set) Let X be a set. Then X is a Tarski Infinite Set if there exists a nonempty S ( P(X) such that for all Y ∈S there exists Z ∈S such that Y ( Z. In some sense, Tarski had appropriately defined what one may want an infinite set to be. We now argue that any Tarski Infinite set must be, informally, an infinite set and conversely. Indeed, let X be a Tarski Infinite Set. Then there exists a nonempty S⊆P(X) such that for all Y ∈ S there exists Z ∈ S such that Y ( Z. Intuitively then, S itself must be “infinite”. To wit, there exist s0,s1,s2, ··· ∈ S such that s0 ( s1 ( s2 ( ··· . Thus we can choose x0 ∈ s0, x1 ∈ s1 \ s0, x2 ∈ s2 \ s1, and so on. It follows that these objects must be distinct. Since x0,x1,x2, ··· ∈ X, we see that X must be “infinite.” Conversely, now let X be (intuitively) an “infinite” set. Since X is “infinite” we can pick x0,x1,x2, ···∈ X such that xi =6 xj for i =6 j. Define S := {{x0}, {x0,x1, }{x0,x1,x2}, ···}. 4 Then S =6 ∅ and S ⊆ P(X) such that for all x ∈ S there exists a y ∈ S such that x ( y. Several years later, John von Neumann introduced the now standard axiom pos- tulating the existence of an infinite set. Before we give von Neumann’s definition, we must define what is contemporarily know as an inductive set. Definition 4. For a set x, let x+ := x ∪ {x}. Now define a set I to be inductive if (1) ∅ ∈ I, and (2) n+ ∈ I whenever n ∈ I. Von Neumann’s Axiom of Infinity postulates the existence of an inductive set. One may argue that any inductive set is informally infinite. Indeed, let I be an in- ductive set. From (1) of the definition, ∅ ∈ I. It follows from (2) that ∅+ ∈ I. Applying (2) again we find that ∅++ ∈ I. Continuing in the fashion we see ∅, ∅+, ∅++, ∅+++, ··· ∈ I. Since these members of I are distinct (this is easy to show) and because we can continue the construction of the above members indefi- nitely, we find that I satisfies our informal notion of an infinite set. There are some sets, however, that appear to be infinite yet fail to be Inductive sets. Indeed, let I be and inductive set and choose y∈ / I. Next, set I′ := {{y, i}: i ∈ I}. It is easy to see that F : I → I′ by F (i) = {y, i} is a bijection between I and I′. Intuitively then, one may feel that I and I′ must be the same “size”, yet we now show that neither properties (1) or (2) from Definition 5 holds for I′. Indeed, ∅ ∈/ I′ as for all i′ ∈ I′, y ∈ i′. Furthermore, for any i′ ∈ I′, i′ contains exactly two sets while 5 i′+ contains exactly three sets. Thus i′+ cannot be in I′. Thus the set I′, although seemingly “infinite”, is not an inductive set. 6 CHAPTER II PRELIMINARIES We now state the standard axioms for ZC − I set theory in which we will be working. We refer the reader to the references for two standard texts on set theory. Axiom 1. (Extensionality Axiom) Let A and B be sets. If for all sets x: x ∈ A if and only if x ∈ B, then A = B . In other words, if two sets have exactly the same members, then they are equal. Axiom 2. (Empty Set Axiom) There exists a set ∅ such that for all sets x, x∈ / ∅. Intuitively, there is a set having no members. Moreover, Extensionality implies that there is a unique set with no members. Axiom 3. (Pairing) Let A and B be sets. There exists a set C such that for all sets x: x ∈ C if and only if x = A or x = B. More informally, for any sets A and B, there is a set having as members just A and B. Axiom 4. (Union Axiom) Let A be a set. Then there is a set B such that for all sets x: x ∈ B if and only if there is some y ∈ A such that x ∈ y. The Axiom of Union asserts that if A is any set, then the members of the members of A form a set.