INITIAL EMBEDDINGS IN THE SURREAL NUMBER TREE A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Elliot Kaplan April 2015 Contents 1 Introduction 1 2 Preliminaries 3 3 The Structure of the Surreal Numbers 11 4 Leaders and Conway Names 24 5 Initial Groups 33 6 Initial Integral Domains 43 7 Open Problems and Closing Remarks 47 1 Introduction The infinitely large and the infinitely small have both been important topics throughout the history of mathematics. In the creation of modern calculus, both Isaac Newton and Gottfried Leibniz relied heavily on the use of infinitesimals. However, in 1887, Georg Cantor attempted to prove that the idea of infinitesimal numbers is a self-contradictory concept. This proof was valid, but its scope was sig- nificantly narrower than Cantor had thought, as the involved assumptions limited the proof's conclusion to only one conception of number. Unfortunately, this false claim that infinitesimal numbers are self-contradictory was disseminated through Bertrand Russell's The Principles of Mathematics and, consequently, was widely believed throughout the first half of the 20th century [6].Of course, the idea of in- finitesimals was never entirely abandoned, but they were effectively banished from calculus until Abraham Robinson's 1960s development of non-standard analysis, a system of mathematics which incorporated infinite and infinitesimal numbers in a mathematically rigorous fashion. One non-Archimedean number system is the system of surreal numbers, a sys- tem first constructed by John Conway in the 1970s [2]. The surreal numbers are of particular interest because they form (in a sense that can be made precise) the largest possible real-closed ordered field in von Neumann-Bernays-G¨odelset the- ory, one of the most widely used set theories in mathematics [4]. Not only is the system of surreal numbers the largest real-closed ordered field, it contains all other real-closed ordered fields as initial subfields [5]. The surreal numbers, No, form both an ordered group and an ordered field 1 Figure 1: The initial stages of the surreal number tree. with a full lexicographically ordered binary tree structure where x <s y is read x is simpler than y and in which sums and products are defined to be the simplest elements consistent with No's ordered group and ordered field structure. In this thesis, I consider a class of s-hierarchical ordered structures, structures first intro- duced in [5] which generalize No's simplicity structure and which are all isomorphic to initial substructures of No. Included in this class are some familiar structures, such as the field of real numbers and the proper class of ordinals. In the following sections, I will present theorems stating precisely which ordered groups and or- dered integral domains are isomorphic to initial subdomains and initial subgroups of this surreal number tree, revealing them to be s-hierarchical ordered structures. This extends the theorem on initial subfields published by Philip Ehrlich in [5]. I 2 will also present some additional results regarding initial subdomains of No. I will begin by introducing some mathematical preliminaries in Section 2, such as sets, functions and the ordinal numbers. I will then introduce the surreal num- bers themselves and some algebraic structures in Section 3. It is in this section that I will introduce the class of s-hierarchical ordered structures. I will discuss Conway Names and how they provide us with a view of the surreal numbers as a vector space over the real numbers in Section 4. I will present the main theo- rems and their proofs in Sections 5 and 6 and I will end with some open questions and closing remarks in Section 7. Throughout, I will use notation and definitions consistent with those used in [5]. Much of the groundwork for this thesis was established in [5]. The theorems in this thesis were formulated jointly with Philip Ehrlich and many of the results were proved with his aid. Portions of this thesis were presented at the Annual North American meeting of the Association for Symbolic Logic on March 25, 2015 at the University of Illinois at Urbana-Champaign. In addition to Dr. Ehrlich, I would like to thank Dr. Todd Eisworth for all of his help over the last four years. 2 Preliminaries 2.1 Sets and proper classes A set is, naively, a collection of objects, a conception that has, since its origin, been made rigorous through various axiomatizations. In this thesis, the underly- ing set theory will be von Neumann-Bernays-G¨odelset theory (NBG). This is a conservative extension of Zermelo-Fraenkel set theory with choice (ZFC), meaning 3 that a statement in the language of ZFC is provable in ZFC if and only if it is provable in NBG. I will not go through most of the axioms of NBG, but I will discuss the notion of a \proper class," a type of entity which is formalized in NBG and only presented informally in ZFC. Originally, set theory was a naive theory, meaning that it was not axiomatized using formal logic but, instead, was defined using natural language. In naive set theory, any definable collection of objects can be considered as a set. For example, the collection of all sets could be considered a set. However, in 1901, Bertrand Russell showed that this naive set theory was self-contradictory as, if any definable collection of objects is a set, then the collection of all sets that are not members of themselves is a set. This set, the Russell set, can be formally written as fx : x2 = xg. The problem arises when we ask: is the Russell set a member of the Russell set? If it is, then it is a member of itself, so it cannot be in the Russell set. However, if it is not, then it is not a member of itself, so it is in the Russell set. There are various axiomatizations of set theory which avoid this paradox, but the method used in NBG set theory is to say that some collections of objects are simply \too big" to be sets. One such object is the collection of all sets. These objects are referred to as proper classes in NBG. All proper classes can be put in a one-to-one correspondence with the proper class of all sets. Two important proper classes in the context of this thesis are the collection of all ordinals, On, and the proper class of surreal numbers, No. Throughout this thesis, the term class will mean a set or a proper class and the term set will be used only when the class in question either is or must be a set. The notation x 2 X means that the object x is an element of the class X. 4 Classes are uniquely defined by their members, meaning that if the set A contains only the member x, written A = fxg, and B = fxg as well, then A = B.A consequence of this is that the set fx; xg is equal to the set fxg (both sets contain exactly the same elements). Another consequence is that there is at most one set which contains no elements. The existence of such a set, called the empty set, is implied by other axioms of NBG. The empty set is referred to by the symbol ;.A class Y will be said to be a subclass of a class X (written Y ⊆ X) if and only if for every y 2 Y , y is also in X. Note that by this definition, X ⊆ X. A class Y is a proper subclass of a class X (Y ( X) if Y is a subclass of X which is not equal to X. This means that there is an element of X which is not in Y . An indexed collection of classes is written as fXγ : γ 2 Γg. This means that for every element γ of the indexing class Γ, there is a corresponding class Xγ in the collection. If this collection both a set and well-ordered, we may write this collection as fXα : α < βg, meaning for every ordinal α less than some ordinal β, Xα is in the collection. If this collection is a well-ordered proper class, we may write the collection as fXα : α 2 Ong, meaning for every ordinal α, Xα is in the collection. The definition of a well-ordering and a construction of the ordinals will S be given later. Given a family of classes, we can construct its union, γ2Γ Xγ and T its intersection γ2Γ Xγ. The union is the class of all elements found in any class in the collection and the intersection is the class of elements found in every class in the collection. The union and intersection of two classes, A and B, are written A [ B and A \ B, respectively. An equivalence relation ∼ on a class A is a two-place relation where, given any a, b and c 2 A: 5 1. a ∼ a (reflexivity). 2. If a ∼ b then b ∼ a (symmetry). 3. If a ∼ b and b ∼ c then a ∼ c (transitivity). Given an equivalence relation ∼ on A, we can define the equivalence class of an element a 2 A, written [a], as the class fb 2 A : a ∼ bg. Given any a; b 2 A, either [a] = [b] or [a] \ [b] = ;, that is, either the equivalence classes of a and b are identical or disjoint.