[...]... segment of a and b where a * b is the element c whose length is the least a such that a(a) * b(a) and such that c(3) = a(3) = b(3) for 3 < a If a = b then c = a = b AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 8 If one of a or b is an initial segment of the other, then c is the shorter element If neither is an initial segment of the other, then either a(y) = + and b(y) = - or a(y) = - and b(y)... us to use the methods of [ 1 ] in dealing with composite operations, as we shall see, for example, in the AN INTRODUCTION TO THE THEORY OF THE SURREAL NUMBERS 16 proof of the associative law for addition Theorem 3.3 The surreal numbers form an Abelian group with respect to addition The empty sequence is the identity, and the inverse is obtained by reversing all signs (Note that one should be aware of. . .AN INTRODUCTION TO THE THEORY OF THE SURREAL NUMBERS 2 In fact, it is because the system seems to be so natural to the author that the first sentence contains the word "discovered" rather than "constructed" or "created." Thus the fact that the system was discovered so recently is somewhat surprising Be that as it may, the pioneering nature of the subject gives any potential reader the opportunity of. .. b e G and a x e F Thus there exists a sequence of length a, such that for all y < a there exist a e F and b e G such that a(3) = d(3) = b(3) for 3 < y AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 6 By hypothesis on a, d cannot be an initial segment of an element in F as well as an element in G Furthermore, an element of F which does not have d as an initial segment must be less than d (Otherwise... implies that b and x a i a i b°a 1 is an increasing function of b iff a < a x and AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 24 the second expression that b°a1 is an increasing function of a iff ab > 1 Hence b°ax is an increasing function of one of the variables iff the other variable is upper At the same time the function preserves sides iff the fixed variable is upper All this can be unified... in the proof of theorem 2.1 It can also be seen trivially as follows Suppose c has minuses Let d be the initial segment of c of length Y where Y is the least ordinal at which c has the value plus Then clearly F < d and d has shorter length than c This contradicts the minimality of the length of c Theorem 2.2a If F = < then f > F|G consists solely of minuses Proof Similar to the above Note that the. .. direct limit of objects with respect to a directed set, a cofinal subset gives rise to an isomorphic object Definition (F',G') is cofinal in (VaeF)(3beF')(b>a) A (VaeG)(3beG' (F,G) if AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS It is clear that (F,G) is cofinal in (F,G), and (F",GM) cofinal in (F',G') and (F',G') cofinal in (F,G) (F\G") cofinal in (F,G) Also if F C F 1 and G c G 1 then is cofinal in... in the /a"1" /a"1" = a 1 Note that as in the case of division, what we did was to insert just enough terms into F and G in order to force the betweenness condition Again, just as in the case of division, we have what is needed to prove a uniformity theorem 27 4 REAL NUMBERS AND ORDINALS A INTEGERS The main task of this chapter is to show that the surreal numbers contain both the real numbers and the. .. d We now check the hypotheses of the cofinality theorem The betweenness property of ab follows from the same computation as in the latter part of the proof of theorem 3.4 For example, since ab - f(c',d') = (ab-c'b) - (ad'-c'd 1 ), P(a,c\b,d') says that ab > f(c',d') The other parts of the betweenness property follow the same way Note that we are now going in the reverse direction to the one we went... from the context Cofinality will be used to sharpen theorem 2.4 to obtain the canonical representation of a as F|G Of course, the representation in theorem 2.4 itself may be regarded as the "canonical" representation The choice is simply a matter of taste Theorem 2.8 Let a be a surreal number Suppose that F1 = {b: b < a DEFINITION AND FUNDAMENTAL EXISTENCE THEOREM 11 and b is an initial segment of a} and . b then c = a = b. AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 8 If one of a or b is an initial segment of the other, then c is the shorter element. If neither is an initial segment of the other,. interesting properties. AN INTRODUCTION TO THE THEORY OF THE SURREAL NUMBERS 2 In fact, it is because the system seems to be so natural to the author that the first sentence contains the word "discovered". this book is to give a systematic introduction to the theory of surreal numbers based on foundations that are familiar to most mathematicians. I feel that the surreal numbers form an exciting system