Radboud Universiteit Nijmegen

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1 Radboud Universiteit Nijmegen Faculteit der Natuurwetenschappen, Wiskunde en Informatica Uniqueness of the Hyperreal Field Noud Aldenhoven ( ) Supervisor: dr. W.H.M. Veldman Second reader: prof. dr. A.C.M. van Rooij January 10, 2010

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3 Contents I The Hyperreal Field 9 1 Some basic definitions and facts about Model Theory Structures Elementary Languages on Structures Tarski s truth definition Important definitions Important facts On the Existence of Ultrafilters What is an Ultrafilter? Properties of Ultrafilters Nontrivial Ultrafilters The Hyperreal Field The construction Extending maps and relations on R to R R as an ordered field Infinitesimals in R Uniformly continuous functions on R II Uniqueness 27 4 A small introduction to Set Theory Equinumerosity Order Transfinite induction and recursion Zorn s Lemma Continuum Hypothesis η α -ordering Do there exist η 1 -orderings? A back-and-forth method Uniqueness of R ( R, ) is an η 1 -order Very small course in Field extensions All η 1 -ordered real closed fields of cardinality the same as R are isomorphic A How many ultrafilters? 49 3

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5 Preface This thesis is about Nonstandard analysis and the uniqueness of the Hyperreal Field. What is Nonstandard analysis? Nonstandard analysis is a technique which provides a logical foundation for the idea of infinitesimals. These are for example numbers greater than 0 but smaller then every positive rational number. Newton and Leibniz used infinitesimal methods in their development of the calculus, such as integration and differentiation. However they were unable to make these infinitesimals precise. Weierstrass and others provided the formal epsilon-delta idea of limits in the eighteenth century. Then Robinson developed nonstandard analysis in the sixties. His original approach was based on these non-standard models of the field of real numbers. These numbers are called the Hyperreal Numbers and they form a (non- Archimedean) field, called the Hyperreal Field. This thesis is about the Hyperreal Field. A small overview We can split this thesis into two parts. Part one, which consists of chapters 1, 2 and 3, gives a brief introduction to Non-Standard Analysis and the Hyperreal Field. Part two, which consists of chapters 4 and 5, considers the uniqueness of Hyperreal Field. CH1. We develop some Model Theory in order to understand the Hyperreal Field. We introduce some basic model-theoretic definitions, including Tarski s truth definition. At the end of this chapter we mention important facts. CH2. In this chapter we give the definition of an ultrafilter. We look at more special ultrafilters, called nontrivial ultrafilters, and we prove, using Zorn s Lemma, that they exist. CH3. Here we explain the construction of the hyperreal field, also called the ultrapower of R. It is often written as R. We show that it is elementarily equivalent to (R, <, +,, 0, 1) and that this hyperreal field contains infinitesimals, for instance numbers greater than zero but smaller then any positive rational number. This gives rise to a new view on real analysis. We will conclude with a proof of one of Weierstrass s theorems concerning R by making an excursion to the hyperreal field. CH4. In this chapter we introduce Set Theory in order to describe η α -orders. We first explain when two sets are of the same cardinality. We introduce some order theory. We explain how to define functions by transfinite induction on a given well ordered set. Then we give some equivalents to the axiom of choice and we mention an important consequence of the Continuum Hypothesis. Finally, we define η α -orders, show that there exists η 1 -orders and show some useful facts of these orders. CH5. In the last chapter we we give a conditional proof of the uniqueness of R. We first prove that R is a η 1 -ordering. We then show, under the assumption that the continuum hypothesis is true, that all real-closed fields of cardinality equal to R with an η 1 -ordering are isomorphic. We conclude that R is unique up to isomorphism if (CH) is true. In the appendix of this thesis we calculate the cardinality of the set of all ultrafilters on N. 5

6 Acknowledgement My thesis was supervised by dr. W.H.M. Veldman. I want to thank him for providing me with the very interesting subject and patiently and concisely reading all preliminary versions of this thesis, saving me from numerous mistakes. Also I want to thank him for the many inspiring meetings we had. I want to thank prof. dr. A.C.M. van Rooij for being the second reader and for the many illuminating and pleasant discussions we had. Finally I want to thank Sep for finding some mistakes in the preliminary versions and Sander and Tim for their support. 6

7 Notations & Basic definitions We begin with some useful notation for this thesis. Let X be a set. Let P(X) denote the set of subsets of X. N, Z, Q and R denote the sets of natural numbers, integers, rational numbers and real numbers, respectively. N does contain 0. Let f : X Y be a function. f is called injective iff for all x 0, x 1 f if f(x 0 ) = f(x 1 ) implies x 0 = x 1. A function is called surjective iff for all y Y there exists some x X such that f(x) = y. f is bijective iff it is surjective and injective. The domain of f is denoted by dom(f). Inclusion of sets is denoted by. We reserve for strict inclusion. So A B means that A B and that A B. Let X, Y be sets, f : X Y a map and E X. The restriction f E of the map f is obtained by restricting the domain of f to E, i.e. f E = {(a, b) f : a E}. A map f is called a partial function on a set X to a set Y if its domain is a subset of X and its image a subset of Y. We denote the set of all partial functions from X to Y as P(X; Y ). We write (AC) for the Axiom of Choice, which can be stated as follows: For every set X of non-empty sets there exists a function f such that for every S X, f(s) S. 7

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9 Part I The Hyperreal Field 9

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11 Chapter 1 Some basic definitions and facts about Model Theory We develop some Model Theory in order to understand the Hyperreal Field. We introduce some basic model-theoretic definitions, including Tarski s truth definition. At the end of this chapter we mention some important facts which will be used later. 1.1 Structures We start with the definition of a structure. Definition A structure A = (A, R 0,..., R m, f 0,..., f n, c 0,..., c p ) is given by: 1. A nonempty set A, called the domain of the structure A. 2. A finite sequence R 0,..., R m of relations on the set A; for each 0 i m there exists k i such that R i A ki. 3. A finite sequence f 0,..., f n of operations on the set A; for each 0 j n there exists l j such that f j : A lj A. 4. A finite sequence c 0,..., c p of elements of A called constants. The triple ((k 0,..., k m ), (l 0,..., l n ), p + 1) is called the signature of the structure A. For example, if we take the signature ((2), (2, 2), 2) we find the real field as an example of structure of this signature. Thus, the real field is the structure (R, <, +,, 0, 1), where < is the standard order, + and are addition and multiplication and 0 and 1 are the units for + and. 1.2 Elementary Languages on Structures Let S = ((k 0,..., k m ), (l 0,..., l n ), p + 1) be a signature. Definition We define the first-order language of signature S. This language with equality, which we denote by L, is a set of formulas containing the following symbols: 1. Individual variables: x 0, x 1, x 2, Connectives:,. 3. Quantifier:. 4. Relation symbols, operation symbols and constant symbols: =, R 0,..., R m, f 0,..., f n and c 0,..., c p. 5. Auxiliary symbols (, ), [, ]. 11

12 The collection of these symbols is called the alphabet of L. The collection of the (individual) terms of L is given by the following inductive definition. 1. Each individual variable x i is a term of L. 2. Each individual constant symbol is a term of L. 3. For i < n and for each sequence t 0, t 1,..., t li of terms of L, the sequence f i (t 0, t 1,..., t li ) is a term of L. 4. We obtain all terms of L from finitely many individual variables and individual constant symbols by finitely many applications of step 3. The set of basic formulas of L is given by: 1. For all terms s, t, the expression s = t is a basic formula. 2. For each relation symbol R i and terms t 0,..., t ki, the expression R i (t 0,..., t ki ) is a basic formula. 3. The basic formulas of L are all formulas given by 1 and 2. We introduce the set of formulas of L 1. Every basic formula of L is a formula of L. 2. For all formulas φ, ψ of L, (φ) (ψ) is a formula. 3. For every formula φ of L, (φ) is a formula. 4. For each formula φ and i N, x i [φ] is a formula. 5. We obtain all formulas of L from finitely many basic formulas by finitely many applications of steps 2, 3 and 4. Let φ and ψ be formulas. To simplify some formulas we define φ ψ as ( φ ψ), φ ψ as ( ψ φ) and x[p(x)] as x[ P(x)]. Consider the signature ((2), (2, 2), 2). Let be a relation symbols, and operation symbols and 0 and 1 constants for this signature. Some examples of formulas in the language belonging to this signature are x y[x 1 = y y x], x y[y 0 y y = x], x y[ (x = y) z[(x z z y) (y z z x)]], x[x 0 = 0], 1 y, x[1 1 = x x 0]. However, these symbols does not have any meaning yet. Consider (R, <, +,, 0, 1). The strange symbols in the formulas above can be applied to the real field. Remark that if we do so not all formulas seem to be true in the structure (R, <, +,, 0, 1). There are even formulas for which not all variables are defined. Before we explain when a formula is said to be true or not in a given structure, we create a function that calculates all undefined variables. Definition Let φ L a formula. We define FVar : L P({x 0, x 1, x 2,...}) as follows, by induction. 1. For each variable x i of L, FVar(x i ) = {x i }. 2. For each constant c i of L, FVar(c i ) =. 3. For each relation R i and terms t 0,..., t ki of L, FVar(R i (t 0,..., t ki )) = j k i FVar(t j ). 12

13 4. For each function f j and terms t 0,..., t lj, t lj+1 of L, FVar(f j (t 0,..., t lj ) = t lj+1) = i l FVar(t j+1 i). 5. For formulas φ, ψ of L, FVar((φ) (ψ)) = FVar(φ) FVar(ψ), FVar( (φ)) = FVar(φ), FVar( x i [φ]) = FVar(φ)\{x i }. Definition We write φ = φ(x 0, x 1,..., x n ) for a formula φ such that FVar(φ) {x 0,..., x n }. A formula φ of L is called a closed formula if FVar(φ) =. Example The formula φ(y) = y > 1 is a formula with one variable, since FVar(φ) = {y}. The formula ψ = x[1 1 = x x 0] is a closed formula since FVar(ψ) = FVar(1 1 = x x 0)\{x} = (FVar(1 1 = x) FVar(x 0))\{x} = {x}\{x} =. 1.3 Tarski s truth definition Let A = (A, R 0,..., R m, f 0,..., f n, c 0,..., c p ) be a structure and S = ((k 0,..., k m ), (l 0,..., l n ), p + 1) the signature of A. Let L be the first-order language of signature S, where we use the following (arbitrary) symbols: R 0,..., R m relation symbols, f 0,..., f n operation symbols and c 0,..., c p constant symbols. Given a formula φ = φ(y) of the language of signature S. We want to say something like: if we choose the value 37 of A for the variable y the formula holds in the structure A. Definition Let t = t(x 0,..., x n ) be a term in L and (a 0,..., a n ) a sequence of elements of A. We define the evaluation of t at (a 0, a 1,..., a n ) with induction, 1. For each i n and variable x i of L, x i [a 0,..., a n ] = a i. 2. For each constant c i of L, c i [a 0,..., a n ] = c i. 3. If t has the form f j (t 0,..., t li ), then f j (t 0,..., t lj )[a 0,..., a n ] = f i (t 0 [a 0,..., a n ],..., t lj [a 0,..., a n ]). We are ready to define when a formula is true in a structure. Definition (Tarski s truth definition) Let φ = φ(x 0,..., x n ) be a formula of L, (a 0,..., a n ) a sequence of elements of A. Inductively, we say that A realizes φ in (a 0,..., a n ), which we denote by if and only if A = φ[a 0,..., a n ], 1. If φ is of the form (s) = (t) where s and t are terms of L, then A = ((s) = (t)) if and only if s[a 0,..., a n ] = t[a 0,..., a n ]. 2. For all terms t 0,..., t ki, if φ is of the form R i (t 0, t 1,..., t ki ) then A = R i (t 0,..., t ki )[a 0,..., a n ] if and only if (t 0 [a 0,..., a n ],..., t ki [a 0,..., a n ]) R i. 3. For all formulas ψ and θ, if φ is of the form (ψ) (θ) then A = ((ψ) (θ))[a 0,..., a n ] if and only if A = (ψ)[a 0,..., a n ] and A = (θ)[a 0,..., a n ]. 4. If φ is of the form (ψ), then A = (ψ)[a 0,..., a n ] if and only if not A = ψ[a 0,..., a n ]. 5. If φ is of the form x i [(ψ)], relabeling the variables we may assume x i = x n+1, then A = ( x n+1 [(ψ)])[a 0,..., a n ] if and only if there exists a A such that A = (φ)[a 0,..., a n, a]. 13

14 1.4 Important definitions Take signature S = ((k 0,..., k m ), (l 0,..., l n ), p + 1). Let A = (A, R 0,..., R m, f 0,..., f n, c 0,..., c p ) and B = (B, S 0,..., S m, g 0,..., g n, d 0,..., d p ) be structures of this signature. Definition f is a monomorphism from A to B iff 1. f is an injective function from A to B. 2. For each i n, let (a 0,..., a ki ) be a sequence of elements of A. Then (a 0,..., a ki ) R i iff (f(a 0 ),..., f(a ki )) S i. 3. For each i m, let (a 0,..., a li ) be a sequence of elements of A. Then f(f j (a 0,..., a lj )) = g j (f(a 0 ),..., f(a lj )). 4. For all k p, f(c k ) = d k. f is an isomorphism from A to B iff f is a monomorphism from A to B and is surjective as a map from A to B. Definition Let f be a monomorphism from A to B, f is called an elementary monomorphism if and only if for every formula φ = φ(x 0,..., x n ) and for every sequence (a 0,..., a n ) of elements of A, A = φ[a 0,..., a n ] iff B = φ[f(a 0 ),..., f(a n )]. If there exists an elementary monomorphism from A to B then we write A B. Definition A is elementarily equivalent to B if for every closed formula φ, A = φ iff B = φ. Notation: A B For example, (Q, <) is elementarily equivalent to (R, <). On the other hand φ = x[0 x y[y y = x]] is true in (R, <, +,, 0, 1) but not in (Q, <, +,, 0, 1), so the structures (R, <, +,, 0, 1) and (Q, <, +,, 0, 1) are not elementarily equivalent. 1.5 Important facts Definition We say that A realizes the formula φ of L if there is an sequence (a 0, a 1,..., a n ) from A such that A = φ[a 0, a 1,..., a n ]. Definition Let Γ be a set of closed formulas in L. A realizes Γ if A realizes every formula of Γ. Notation: A = Γ. Example Here are a set of closed formulas of the language of signature ((2), (2, 2), 2). This set is called ORDFIELD() and characterizes ordered fields. x[ (x x)] x y z[(x y y z) x z] x y[x = y x y y x] x y z[(x y) z = x (y z)] x y[x y = y x] x[x 0 = x] x y[x y = 0] x y z[(x y) z = x (y z)] x y[x y = y x] x[ (x = 0) x 1 = x] x[ (x = 0) y[x y = 1]] x y z[x (y z) = (x y) (x z)] } } x y z[x y x z y z] x y[(0 x 0 y) (0 x y)] is a strict total ordering, (A, ) forms an Abelian group, (A,, ) forms an Abelian ring, and are distributive, A is a totally ordered field. We see that (R, <, +,, 0, 1) = ORDFIELD(). So (R, <, +,, 0, 1) is an ordered field. 14

15 We end this chapter with a theorem from model theory. Theorem (Compactness theorem for first-order-logic) Let Γ be a set of formulas in L. If every finite subset of Γ has a structure A such that A realizes this finite set of formulas, then there exists a structure B such that B realizes Γ. We will not prove the Compactness theorem here; for a detailed proof, see [CK73] pp

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17 Chapter 2 On the Existence of Ultrafilters In this chapter we give the definition of an ultrafilter. In different branches of mathematics, different definitions of ultrafilters occur. We give one that is most common in set theory. In the following sections of this chapter we consider more special ultrafilters, called nontrivial ultrafilters, and we prove, using Zorn s Lemma, that they always exist. 2.1 What is an Ultrafilter? Definition Let X be a nonempty set. Then F P(X) is called a filter on X iff i. / F, X F, ii. for all A, B F, A B F, iii. for all A F and B P(X), if A B, then B F. Example Let X be a non-empty set and Y X such that Y. Then F = {Z P(X) : Y Z} is a filter. Its properties are easy to check and are left to the reader. We want to have an extra property on filters. This gives the following definition. Definition Let X be a nonempty set, then F P(X) is called an ultrafilter on X if 1. F is a filter, 2. Let Y X, then Y F or X\Y F, but not both. We could ask ourselves if, given a set X, there exists an ultrafilter on X. Our question is answered by the following example. Example Let X be a set and x 0 X. Then the subset of P(X) F = {U P(X) : x 0 U}, is an ultrafilter. One verifies this immediately, as follows: 1.i. / F; otherwise x 0. And because x 0 X we have X F. 1.ii. Let A, B F. Then x 0 A and x 0 B, so x 0 A B and therefore A B F. 1.iii. Let A F and B P(X). If A B and x 0 A, then certainly x 0 B, so B F. 2. Let Y X, then x 0 Y or x 0 X\Y, so X F or X\Y F. We cannot have both, otherwise we would have x 0 Y X\Y =, which is absurd. This ultrafilter shown in example is not very interesting, it is not useful for the construction of non-standard analysis. Definition Let X be a nonempty set, F P(X) is called a trivial ultrafilter if it is of the form of example An ultrafilter which is not of this form is called a nontrivial ultrafilter. 17

18 2.2 Properties of Ultrafilters Theorem Let F be an ultrafilter on a nonempty set X. There exists A F such that A is finite, if and only if F is a trivial ultrafilter. Proof. If there is a A such that {a} F, then F is trivial. So it is sufficient to show that there exists a X such that {a} F. Let A = {a 1, a 2,..., a n } and suppose that for every a i A, {a i } / F. Because F is an ultrafilter we have that X\{a i } F. So A X\{a 1 } X\{a 2 }... X\{a n } = A X\{a 1, a 2,..., a n } = F. Which is nonsense. Therefore we have that there is a i A so that {a i } F. If F is trivial then obvious F contains a finite set. Ultrafilters can be defined in different ways, we give some important equivalent definitions of ultrafilters. Theorem Let X be a nonempty set, F a filter on X. Then the following statements are equivalent, i. F is an ultrafilter. ii. If F is a filter and F F, then F = F. (so F is a maximal filter on X) iii. If A B F then A F or B F. Proof. i. ii. Suppose that F is an ultrafilter. Let F be a filter on X such that F F and suppose F F. Then there is A F for which A / F thus X\A F. Because F F we know that X\A F and, because F is a filter, A X\A = F. By contradiction we have F = F. ii. i. Suppose that if F is a filter on X and F F, then F = F. We want to show that F is an ultrafilter. Because F is already a filter, it suffices to prove that clause 2 holds. I.e. for all Y X, Y F or X\Y F. Let A X and suppose A / F. We want to show that X\A F. Define G = {B X : A B F}. It is easy to show that G is a filter and F G, but because F is a maximal filter, F = G. Then so X\A G, and therefore X\A F. A (X\A) = X F, i. iii. Suppose that F is an ultrafilter. Let A, B P(X), A B F, but, towards contradiction, suppose that A, B / F. Then X\A, X\B F, so, since F is a filter, X\A X\B = X\(A B) F but then which is absurd. = X\(A B) (A B) F, iii. i. For each subset A of X, X\A A = X F. So X\A F or A F. They cannot be both in F since then = X\A A F. 18

19 2.3 Nontrivial Ultrafilters In this section we show that using Zorn s Lemma we can always find a nontrivial ultrafilter on a given set X. If you do not know what Zorn s Lemma is you can first read the part about Zorn s Lemma in chapter 4. Theorem (Ultrafilter theorem) Let X be a nonempty set, F a filter on X. Then there exists an ultrafilter F on X such that F F. Proof. This proof uses Zorn s Lemma. Let F be a filter on X. Define G = {G P(X) : F G and G is a filter} We want to show that G has a maximal element F which is a nontrivial ultrafilter. Let D G, where D is linearly ordered under inclusion. Let E = D. D D It is straighforward to see that E is a filter and F E. Therefore E is an upper bound of D in G. Hence by Zorn s Lemma, G has a maximal element F. Since F G we have F F and for every filter F on X, if F F, then F G, so that F = F. The Ultrafilter theorem is weaker then Zorn s Lemma in the context of Zermelo-Fraenkel set theory without the axiom of choice (ZF). In (ZF) we can prove the Ultrafilter theorem from Zorn s lemma, but we cannot prove Zorn s Lemma from (ZF) and the Ultrafilter theorem. For more detail see [RR85], DAL 1 p Theorem Let X be a nonempty infinite set. There exists a nontrivial ultrafilter on X. Proof. Let F be the Fréchet filter of X. That is, F = {U P(X) : X\U is finite}. It is straightforward to see that F is a filter. Using theorem we find an ultrafilter F such that F F which is not trivial. So F is a nontrivial ultrafilter on X. Corollary There exists a nontrivial ultrafilter F on N. 19

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21 Chapter 3 The Hyperreal Field In this chapter we explain the construction of the hyperreal field, also called the ultraproduct of R. We show that it is elementarily equivalent to (R, <, +,, 0, 1) and we show that this hyperreal field contains infinitesimals, for instance numbers greater than zero but smaller then any positive rational number. This gives rise to a new view on real analysis. We will conclude with a proof of one of Weierstrass s theorems concerning R by making an excursion to the hyperreal field, saying that every continuous function on a compact interval of R is uniformly continuous. One could ask if there exists a field with infinitesimals. We create one using the compactness theorem We introduce a new individual constant c and take the following set of formulas n times { }} { Γ = ORDFIELD() {0 c} { c... c < 1 : n N}. Now for any finite subset of of Γ there exists a structure which realizes, because for a large integer m we have (R, <, +,, 0, 1, 1 m ) =. This fits the requirements for the compactness theorem, so there exists a structure A which A = Γ. So A is a field and it contains infinitesimals. We show an other construction for a field with infinitesimals. 3.1 The construction Nonstandard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. Robinson s original approach was based on non-standard models of the field of real numbers, which we show here. He wrote a book on this subject and published it in 1966, see [Rob66]. We start from the structure (R, <, +,, 0, 1). We first show how to create the numbers in the hyperreal field. These numbers will be constructed of sequences of real numbers. This gives the following definition. Definition We introduce R N = {f : f : N R} = {(x 0, x 1, x 2,...) : x i R}. Definition On the set R N we define operators + and. For all f, g R N we define, Constant elements in this structure are f + g = n (f(n) + g(n)), f g = n (f(n) g(n)). 0 = n 0 (= (0, 0, 0,...)), 1 = n 1 (= (1, 1, 1,...)). 21

22 For completeness we show the inverse functions for + and, where for all n N g(n) = f = n f(n), f 1 = n g(n), { (f(n)) 1 if f(n) 0 0 otherwise Later on, as exercise, you can check that these inverses are correct. We now have a structure (R N, +,, 0, 1). However this is not a field, since it contains zero divisors. For example the sequences (0, 1, 0, 1, 0, 1,...) and (1, 0, 1, 0, 1, 0,...). Because (0, 1, 0, 1, 0, 1,...) (1, 0, 1, 0, 1, 0,...) = (0, 0, 0, 0, 0, 0,...). To avoid such problems one of these sequences should be called the same as (0, 0, 0,...). This can be done by modding them out to an equivalence relation. For this we first need an equivalence relation, this is were ultrafilters come in play. Definition Let F be a non-trivial ultrafilter on N. We define a relation = F on the set R N as follows. For all f, g R N, f = F g iff {n N : f(n) = g(n)} F. Lemma Let F be a non-trivial ultrafilter on N. Then = F is an equivalence relation on R N. Proof. Let f, g, h R N. Then 1. f = F f, because {n N : f(n) = f(n)} = N F. 2. If f = F g, then {n N : f(n) = g(n)} F. Hence {n N : g(n) = f(n)} F, so g = F f. 3. If f = F g and g = F h, then A = {n N : f(n) = g(n)} F and B = {n N : g(n) = h(n)} F. So {n N : f(n) = g(n) and g(n) = h(n)} = A B F and {n N : f(n) = h(n)} {n N : f(n) = g(n) and g(n) = h(n)}, hence, because F is an (ultra)filter, {n N : f(n) = h(n)} F, so f = F h. If f R N we write [f] for the set of all g R N such that f = F g. Knowing this we are ready to construct the hyperreal numbers. Definition Let F P(X) be a non-trivial ultrafilter. We define the hyperreal numbers with equivalence relation = F, R F = R N /F = {[f] : f R N }. We now define operators + F and F on R F and constant symbols 0 F and 1 F as follows. For all f, g R N [f] + F [g] = [f + g], [f] F [g] = [f g], 0 F = [0] (= [(0, 0, 0,...)]), 1 F = [1] (= [(1, 1, 1,...)]). Note that if f = F f then f + g = F f + g and f g = F f g. So we may conclude that these operations are well-defined on R F. Now ( R F, +,, 0, 1) is a field, as we will prove in the next session. 22

23 Definition Let < be the usual order on R. We define a new order < F on R. For all f, g R N F, Again this operator is well-defined on R F. [f] < F [g] iff {n N : f(n) < g(n)} F. In the next sections we shall see that the structure ( R F, < F, + F, F, 0 F, 1 F) is a field ( the Hyperreal Field) which contains infinitesimals and is elemenarily equivalent to the real field. The next two chapters will explain that this hyperreal field is not isomorphic to the real field. If we had taken a trivial ultrafilter G we would have that ( R G, < G, + G, G, 0 G, 1 G) is isomophic to the real numbers, and this field does not contain infinitesimals. It might be a good exercise to prove that they are isomorphic indeed. 3.2 Extending maps and relations on R to R Given a map φ from R to R or a relation R on R n, we want to extend this map in a natural way to a map φ F from R F to R F and extend this relation R to a relation R F on R n F. In this way we create a method to do real analysis on the Hyperreal Field. A relation R on R n can be extended in the following way. Definition Let R be a relation on R n. We extend R to R F by defining for all x 0,..., x n 1 R N ([x 0 ],..., [x n 1 ]) R F iff {i N : (x 0 [i],..., x n 1 [i]) R} F. Definition Let X R. Then define X F R F by, X F = {[f] : f R N and {n N : f(n) X} F}. Let f, g R N ; if [f] = [g] then {n N : f(n) X} F if and only if {n N : g(n) X} F. So X F is well-defined. A function from X to R can be extended in the following way. Definition Let X R, φ : X R a map, we extend φ to a non-standard map φ : X R. For all f X N φ F ([f]) = [g], where for all n N g(n) = { (φ f)(n) if f(n) X 0 otherwise At last, if x R we extend it as follow to a hyperreal element. Definition Let x R. We define x F = [(x, x, x,...)] F the be the equivalent class of the constant sequence of x. 3.3 R as an ordered field From now on we fix our nontrivial ultrafilter F and simplify our notations from R F, = F, R F, f F, c F to R, =, R, f, c. So we simplify < F, + F, F, 0 F, 1 F to <, +,, 0, 1. In chapter 1 we developed a strong method to prove that a model is a field. We use that one to prove that R is a field. First we prove that (R, <, +,, 0, 1) and ( R, <,, 0, 1) are elementarily equivalent. This follows from a fundamental theorem from model theory. 23

24 Theorem (Ultraproduct theorem, J. Loš, Hyperreal Field case) Let F be a nontrivial ultrafilter. Let R the Hyperreal Field created from F. Let R 0, R 1,..., R m be relations on R, f 0, f 1,..., f n functions on R and c 0, c 1,..., c p constants of R as defined in chapter 1. Construct R 0, R 1,..., R m, f 0, f 1,..., f n, c 0, c 1,..., c p as described in the previous section. For every formula φ = φ(x 0, x 1,..., x m ), for every finite sequence g 0, g 1,..., g m R N, if and only if ( R, R 0,..., R m, f 0,..., f n, c 0,..., c 1 ) = φ[[g 0 ],..., [g m ]] {n N : (R, R 0,..., R m, f 0,..., f n, c 0,..., c p ) = φ[g 0 (n),..., g m (n)} F. This theorem is proved by induction on the structure of φ. For a detailed proof see [CK73], pp Corollary (Transfer Principle) R as field is elementarily equivalent to R as field, i.e. (R, <, +,, 0, 1) ( R, <, +,, 0, 1). Proof. This follows directly from the Ultraproduct theorem. Hence every first order theorem which is true in the real numbers is true in the hyperreals and vice versa. Now it is easy to prove that R is a totally ordered field. Corollary The structure ( R, <, +,, 0, 1) is a totally ordered field. Proof. Since R is a totally ordered field we know that ORDFIELD() is true in R, i.e. (R, <, +,, 0, 1) = ORDFIELD(). From corollary we have that ( R, <, +,, 0, 1) = ORDFIELD(), from which we conclude that ( R, <, +,, 0, 1) is a totally ordered field. Later on in this chapter we make more use of this transfer principle. This concludes the construction of the Hyperreal Field. The Hyperreal Field is different from the Real Field in a sense that they are not isomorphic. In the next three sections we show some applications of the hyperreal numbers. 3.4 Infinitesimals in R It is easy to see there there exist infinitesimals in R. For example, take the following element ǫ = [(1, 1 2, 1 4, 1 8,..., 1 2 n,...)]. Now ǫ is greater then 0 = [(0, 0, 0,...)] but smaller then every positive rational number q = [(q, q, q,...)]. We develop some more theory about infinitesimals. Let x R, we write x for the absolute value of x. Definition We define F = {x R : x < n n N} as the set of all finite elements of R. And we define I = {x R : x < ( 1 n ) n N} as the set of all infinitesimals. We observe that (F, +,, 0, 1) is a ring. Theorem The set of infinitesimals I of R is an ideal in ring F. Proof. Let x, y I and z F, we have x + y I, since for all n N, x + y < ( 1 2n ) + ( 1 2n ) = ( 1 n ). xz I. Because z F, we have some m N, such that z m. So for all n N, xz = x z < ( 1 nm m) = ( 1 n ). 24

25 Definition Let x, y R, we write x y iff x y I. Theorem is an equivalence relation on F. Proof. Let x, y, z F, then 1. x x iff x x = 0 F, which is true. 2. Suppose x y and y z, then x y, y z I. Since I is an ideal, x z = x y + y z = (x y) + (x z) I. 3. Suppose x y, then x y I. So ( 1)(x y) = y x I, and therefore y x. We should read this as follows, if x y then x lies infinitely close to y. It also gives the following homomorphism. Definition We write for the natural homomorphism : F F/I. Theorem Let x R be a finite element. Then there is r R such that x = r. In other words F/I R Proof. Let x F, define A = {r R : r x}. Because x is finite, there exists n R so that x n and so A n. Therefore the supremum of A exists in R. Let sup(a) = r R. For all ǫ > 0, r + ǫ / A and r ǫ A, thus r ǫ x r + ǫ or equivalent x r ǫ. So x r I iff x r iff x = r. This last theorem says that for every finite element x R we can find y R, such that x y. 3.5 Uniformly continuous functions on R Here we show an alternative proof of Weierstrass theorem using non-standard analysis that arrises from the previous sections. Lemma Let f : [a, b] R. f is continuous at x 0 [a, b] iff for all x [a, b], x x 0 f(x) f( x 0 ). (3.1) Proof. Let f : [a, b] R be a continuous function in x 0 [a, b]. Then for all ǫ > 0 there is a δ > 0 such that, R = x dom(f) [ x x 0 < δ f(x) f(x 0 ) < ǫ]. We can transfer this formula to our Hyperreal Field, R = x dom(f) [ x x 0 < δ f(x) f( x 0 ) < ǫ]. Now let x dom(f), x x 0. Then certainly x x 0 < δ, so and because this is true for all ǫ R + we see that f(x) f( x 0 ) < ǫ f(x) f( x 0 ). Conversely, suppose that if x x 0, then f(x) f( x 0 ). Let ǫ R + and take δ > 0 such that δ 0, then Therefore, R = δ > 0 x dom(f) [ x x 0 < δ f(x) f( x 0 ) < ǫ]. R = δ > 0 x dom(f) [ x x 0 < δ f(x) f(x 0 ) < ǫ], which is the standard definition of being continuous at x 0. 25

26 Lemma Let f : [a, b] R. f is uniformly continuous on [a, b] iff for all x, x [a, b], x x f(x) f(x ). (3.2) Proof. Suppose f : [a, b] R uniformly continuous on [a, b]. Then for every ǫ > 0, there exists a δ > 0 such that, R = x dom(f) x dom(f) [ x x < δ f(x) f(x ) < ǫ]. Transferring this formula to the hyperreal structure gives R = x dom(f) x dom(f) [ x x < δ f(x) f(x ) < ǫ]. Suppose x, x R, x x. Then certainly x x < δ and so f(x) f(x ) < ǫ. Because this is true for all ǫ > 0, we have f(x) f(x ). Conversely, suppose that if x x then f(x) f(x ). Let ǫ R +, we can choose an infinitesimal δ R +, so that R = δ x dom( f) x dom( f) [ x x < δ f(x) f(x ) < ǫ]. Using the transfer principle again this gives us: R = δ x dom(f) x dom(f) [ x x < δ f(x) f(x ) < ǫ], which gives us a good δ for each ǫ. With these two lemmas Weierstrass theorem becomes quite elegant. Theorem (Weierstrass) Let f : [a, b] R be continuous, then f is uniformly continuous. Proof. Let x, x [a, b], such that x x. Due to lemma 3.5.2, it is sufficient to show that f(x) f(x ). Remark that since [a, b] is bounded, there is r [a, b] such that x r and x r. Because f is continuous at r, f(x) f( r) and f(x ) f( r). Therefore f(x) f(x ). This is where we stop developing more analysis over the hyperreal field. In the next part of this thesis we consider the hyperreal field from a set theoretical point of view. So given two different (nontrivial) ultrafilters F and G, van we prove that From corollary it follows that ( R F, < F, + F, F, 0 F, 1 F ) ( R G, < G, + G, G, 0 G, 1 G )? (3.3) ( R F, < F, + F, F, 0 F, 1 F ) ( R G, < G, + G, G, 0 G, 1 G ) but from this we are unable to conclude that the structures are isomorphic. It will turn out that, under some strong assumptions, (3.3) is true. One might asks himself or herself if all non-trivial ultrafilters on N are just permutations of each other? In other words: let F and G be two non-trivial ultrafilters on N. Can we find a permutations σ : N N such that F = {σ(x) : X G} = σ(g)? In the appendix we show that there are more ultrafilters than permutations on N. So there are ultrafilters which are not permutations of each other. 26

27 Part II Uniqueness 27

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29 Chapter 4 A small introduction to Set Theory Our goal in this chapter is to introduce sufficient Set Theory in order to describe η α -orders. We first explain when two sets are of the same cardinality. We introduce some order theory. We explain how to define functions by transfinite induction on a given well ordered set. Then we give some equivalents to the axiom of choice and we mention an important consequence of the Continuum Hypothesis. Finally we define η α -orders, show that there exists η 1 -orders and show some useful facts of these orders. Our exposition is partly based on [Mos94]. 4.1 Equinumerosity We start with a fundamental theorem from Set Theory. Definition Two sets A, B are equipollent or equal in cardinality if there exists a bijection between A and B. Notation: A B. Definition Let A and B two sets, then A is lesser than or equal to B if there exists an injective function from A to B. Notation: A c B. Theorem (Cantor-Schröder-Bernstein) For any two sets A and B, if A c B and B c A then A B. Proof. Let f : A B and g : B A be two injective functions. We define the sets A n and B n recursively A 0 = A, B 0 = B, A n+1 = (g f)(a n ), B n+1 = (f g)(b n ). By induction we see that A n g(b n ) A n+1 and B n f(a n ) B n+1. We now define the intersections This gives A = A n, B = B n. n=0 n=0 n=0 n=0 B = B n f(a n ) B n+1 = B. And since f is an injection and f(a ) = f( n=0 A n) = n=0 f(a n) = B we have that f is a bijection from A to B. Also n=0 A = A (A 0 \g(b 0 )) (g(b 0 )\A 1 )..., B = B (B 0 \f(a 0 )) (f(a 0 )\B 1 )... 29

30 Because f and g are injective functions, for every n N, f(a n \g(b n )) = f(a n )\(f g)(b n ) = f(a n )\B n+1, g(b n \f(a n )) = g(b n )\(g f)(a n ) = f(b n )\A n+1. This gives us the following bijection π : A B, { f(x) if n : x An \f(b π(x) = n ) or x A g 1 (x) otherwise Also see figure 4.1. This shows that A B. A A 0 \g(b 0 ) g(b 0 )\A 1 f g 1 B 0 \f(a 0 ) f(a 0 )\B 1 B A 1 \g(b 1 ) g(b 1 )\A 2 f g 1 B 1 \f(a 1 ) f(a 1 )\B 2... A f B Figure 4.1: Proof of Cantor-Schröder-Bernstein s theorem. With this powerful theorem it is not that hard to show that R R. We start with three lemmas. Lemma R (0, 1). Proof. Take bijective map f : R (0, 1) : x arctan(x) π Lemma R N R. Proof. We use Cantor-Schröder-Bernstein s theorem to prove this. c ) Let f : R R N : r (r, r, r,...). This is clearly an injective map, so R c R N. c ) We know that R (0, 1), so we only have to show that there exists an injective map g : (0, 1) N (0, 1). Let (r 0, r 1, r 2,...) (0, 1) N. We write the decimal representations of all r i. If we have to choose between a sequence of infinitely many nines or zeros we choose the zeros. So we take instead of This gives r 0 = 0. q 0 0, q0 1, q0 2, q0 3,... r 1 = 0. q 1 0, q1 1, q1 2, q1 3,... r 2 = 0. q 2 0, q 2 1, q 2 2, q 2 3,... r 3 = 0. q 3 0, q 3 1, q 3 2, q 3 3,

31 Then we use Cantor s famous argument to define g(r 0, r 1, r 2,...) = 0.q 0 0 q1 0 q0 1 q2 0 q1 1 q0 2 q3 0 q2 1 q1 2 q Since r i (0, 1) we have an injective map from (0, 1) N to (0, 1). So R N c R. According to Cantor-Schröder-Berstein s theorem there exists a bijective map between (0, 1) N and (0, 1). Therefore R N R. The following result is more difficult. Theorem Let F be an ultrafilter on N. Then R F R. Proof. We use the Axiom of Choice (AC) here and (again) we will use Cantor-Schröder-Bernstein s theorem to show the presence of a bijection. c ) This is easy, map f(r) = [(r, r, r,...)] is injective. So R c R F. c ) From lemma we know that there exists a bijection φ : R N R. We use (AC) for a choice function ψ : R F R N which chooses for every equivalence class in R F a representative in R N. We define an injective map, g : R F R : x (φ ψ)( x). So R F c R. Therefore we have that R F R. We immediately see that if F and G are two ultrafilters on N than R F R R G. 4.2 Order This section introduces some basic order theory. Definition Let P be a set. An order of P is a binary relation P P such that, for all x, y, z P: i. x x, ii. if x y and y z, then x z, iii. if x y and y x, then x = y. These conditions are named (respectively) reflexivity, transitivity and anti-symmetry. The set P equiped with the order is called a poset, notation: (P, ). If we also have the condition that for all x, y P: iv. x y or y x, we call a total ordering on P and (P, ) a totally ordered set. We define for all x, y P, x < y iff x y and x y. Definition Let P be a set and let be a partial ordering of P. We say that m P is a maximal element of P iff there does not exist n P such that m < n. Definition Let P be a set and a total-order of P. We say that is a wellordering of P iff every non-empty subset of P has a smallest element. The structure (P, ) is then called a well-ordered set. We always write 0 for the smallest element of P. Example The natural numbers N with the natural order are a well-ordered set. Definition Let P be a set and an order of P. Let S : P P a map. S is called a successor function iff for all x P 1. x < S(x), 31

32 2. there is no y P such that x < y < S(x). Notice that in a well-ordered set we can always define a successor function. Lemma Let be a wellordering of P. Then for all x P. If x is not a maximal element then is a successor function on P. S(x) = min{y P : x y} This makes well-ordered sets extremely useful, as we shall see in the next sections. Definition Let P and Q sets. Let P be a partial order of P, Q a partial order of Q and π : P Q. We say that π is an order-embedding iff for all x, y P x P y iff π(x) Q π(y). If π is bijective and an order-embedding we call π an order-isomorphism. Definition Let P be a set and a partial order of P. C P is called a chain iff for all x, y C we have x y or y x. Definition Let P be a set and a partial order of P. Let Q P. m P is called an upperbound of Q iff for all q Q q m. 4.3 Transfinite induction and recursion The following theorem proves the idea behind transfinite induction. Theorem (Transfinite Induction) Let U a set. Let be a wellordering of U. For every property P U, x[ y[(y x P(y))] P(x)] z[p(z)]. Proof. Suppose the opposite, so x[ y[(y x P(y))] P(x)] and z[ P(z)]. Now let y = min{x U : P(z)}. Since z[ P(z)], this y is defined, so P(y ) and y[y y P(y)]. However we also know that x[ y[(y x P(y))] P(x)], thus P(y ). This contradicts the choice of y. With transfinite induction we construct maps by using transfinite recursion. This gives the following theorem. Recall that P(U; E) is the set of all partial functions from U to E. Theorem (Transfinite Recursion) Let U and E two sets. Suppose that is a wellordering of U and h : P(U; E) E. Then for every t U there exists exactly one function which for all u t satisfies the identity σ : {u U : u t} E, σ(u) = h(σ {v U : v u t}). Moreover, if h : P(U; E) E, there exists exactly one map σ such that for all u U σ(u) = h(σ {v U : v u}). Proof. We first show that if such a σ exists than it is unique. Suppose t U and we have distinct σ and σ both satisfying the identity above. Then there is a y t such that σ(y) σ (y). Take the smallest y t for which σ(y) σ (y). This leads to a contradiction, because σ(y) = h(σ {z U : z y} = h(σ {z U : z y}) = σ (y). 32

33 So σ is unique. Next we prove that for each t U there always exists at least one σ with this property. This is proved by transfinite induction. Assume that for each u t there exists exactly one function such that for all v u, σ u : {v U : v u} E σ u (v) = h(σ u {w U : w v}). (4.1) We want to show that there also exists a suitable function with domain {u U : u t}. If t = 0 we define If there is a v such that t = S(v), we define σ 0 =. σ t = σ v {(v, h(σ v ))}. Now, by induction hypothesis, (4.1) holds for all x v and holds for x = v by definition. If there is no v U such that S(v) = t, we show that {σ u : u t} is a chain under, i.e. if x u v t, then σ u (x) = σ v (x). Assume, towards contradiction, that for some u v t and x u we have that σ u (x) σ v (x). Because U is wellordered under we can find the least x such that σ u (x) σ v (x). Then σ u {w U : w x} = σ v {w U : w x}, and therefore which contradicts the choice of x. Now let σ u (x) = h(σ u {w U : w x}) = h(σ v {w U : w x}) = σ v (x) σ t = u t σ u. Because {σ u : u t} is a chain we have that this σ t is a function with domain {w U : w t}, and it satisfies (4.1), since for each x t, there is some u U such that x u t and σ t (x) = σ u (x) This completes the first part of the theorem. = h(σ u {w U : w x}) = h(σ t {w U : w x}). The last part of the theorem follows from the first part. Let U { } be the set U { } such that for all u U, u. This ordered set U { } is again wellordered. Now let h : P(U; E) E. We extend h to h : P(U { }; E) E by { h h(σ) if Domain(σ) U (σ) = e if Domain(σ) for some arbitrary element e E. Now, by applying the first part, we find an unique f : {v U : v } E such that f(x) = h (f {v x}) for all x. But then x U and f : U E, so f(x) = h(f {v x}) for all x U. This theorem is a very powerful tool and will be used throughout the text below. 33

34 4.4 Zorn s Lemma As is well known the axiom of choice implies Zorn s Lemma; it is even equivalent to Zorn s Lemma. We present two related equivalents of the axiom of choice. Theorem Hausdorff s Lemma (HL) Let A be a set and let be a partial ordering of A. Then there exists M A such that (M, ) is a maximal chain in (A, ), i.e. for every a A, if is a total order of M {a} then a M. Theorem Zorn s Lemma (ZL) Let A be a set and let be a partial order of A. If for every chain C A exists an upper bound M A, then there exists a maximal element in (A, ). Theorem Wellordering Theorem (WT) Let A be a non-empty set, then there exists an order of A such that (A, ) is a well ordering. We saw in the previous section that if we can well-order a set, we can apply transfinite induction on this set. So theorem shows that we can always use transfinite induction on given set. We now show that these three statements are equivalent. Theorem The following statements are equivalent: i. Hausdorff s Lemma, ii. Zorn s Lemma, iii. Wellordering Theorem. Proof. i ii. Let (A, ) be a non-empty poset such that for every chain C A we have an upper bound M A. We want to show that (A, ) contains a maximal element. According to (HL) we have a X A such that is a maximal total order of X. Let x be an upper bound of X. Notice that x X because is a total order of X {x} and X is maximal, so X {x} = X. We claim that x is a maximal element of A. Suppose, to get a contradiction, y > x. Then is a total order of X {y} and so: y X and y x, a contradiction. ii iii. Let A be a non-empty set. We will show that there exists an order such that (A, ) is a well-order. We define and introduce an order, B = {(X, X ) : X A and (X, X ) is a well order} (X, X ) (Y, Y ) iff X Y, x X y Y \X : x Y y and X X. (4.2) Now ) is an order of B. Let (X, X ), (Y, Y ), (Z, Z ) B, then a. (X, X ) (X, X ) iff X X, X X = X and X X so (4.2) is obvious true. b. Let (X, X ) (Y, Y ) and (Y, Y ) (Z, Z ) then X Y and Y Z, so X Z. Also X Y Z. And x X y Y \X : x Y y and y Y z Z\Y : y Z z. We conclude that x X z Z\X : x Z z. And therefore (X, X ) (Z, Z ). c. Let (X, X ) (Y, Y ) and (Y, Y ) (X, X ), then X Y and Y X, so X = Y and (X, X ) = (Y, Y ). We want to use (ZL) to find a wellordering on A. Let C B a chain, define D = ( X, X ) = (D, D ). (X, X) C (X, X) C Now D is a well-order of D. It is straightforward to show that D is a total order of D. We show well-order. Let Y D, Y. Then there is Z (X, X) C X such that Y Z. Hence there exists a smallest element m of Y Z Z in the ordering Z. Now suppose that there is n D 34

35 such that n D m. Because Z D we have n Z m. But then n Z. Since Y was arbitrary, every subset of D has a smallest element, so D is a well-order. So D is an upper bound for C. Now from (ZL) it follows that (B, ) contains a maximal element (M, ). We show that M = X. Suppose, toward contradiction, a X but a / M. We create a new chain M {a} such that (M, ) (M {a}, M {a} ) and (M {a}, M {a} ) is a well ordered set, which is in contradiction with the maximality of (M, ). So (X, ) is a well ordered set. iii i. Let A be a non-empty poset with ordering. We wellorder the set A by. Now, by transfinite induction, we define a function φ on A such that for all a A, { {a} if ({a} {φ(b) : b a}, <) is a total ordering φ(a) = otherwise It is straightforward to see that (φ(x), ) is a maximal chain in (X, ). 4.5 Continuum Hypothesis The continuum hypothesis is a hypothesis, advanced by George Cantor in 1877, about the possible size of infinite sets. Definition (Continuum Hypothesis) There is no set of real numbers S R with cardinality intermediate between those of N and R. In logical symbols: X R[X N X R]. We will notate the Continuum Hypothesis as (CH). As is well-known the hypothesis can neither be disproved nor proved from the axioms of Zermelo Fraenkel Set Theory, provided Set Theory is consistent. If we assume that (CH) is true some rather curious theorems appear to be true. Theorem Suppose (CH) then, by the well-ordering theorem (WT), there exists a well-ordering 0 on R such that for every x R is at most countable. {y R : y 0 x} (4.3) Proof. We use (WT) to find a well-order on R. Suppose that there is x R such that {y : y x} is uncountable. Then the following set is defined and not empty, A = {x R : {y : y x} is uncountable }. Since is a well-ordering, we can find a smallest element x 0 A. Now for B = {y R : y x}, we have that B is uncountable and for every b B {y R : y b} is countable. It follows from (CH) that B R. Let φ : R B a bijection. Then 0 is, for every x, y R, defined by x 0 y iff φ(x) φ(y). Now 0 is a well-order of R with the property (4.3). 35

36 4.6 η α -ordering The notation of an η α -ordering was first introduced in the Grundzüge der Mengenlehre, by Felix Hausdorf in 1914, see [Hau14]. Definition Let X be a totally ordered set, r X and A X. We write A < r if for all a A, a < r. If B X we write A < B if for all a A and b B, a < b. In the Grundzüge der Mengenlehre η α -ordering states the following, Definition Let α be an ordinal number. Let L be of cardinality at least ℵ α and a total ordering of L. is called a η α -ordering if and only if: (i) if A, B L, both not empty, of cardinality less than ℵ α, and such that A < B then there is a c L such that A < c < B, (ii) for every non-empty subset D L of power less than ℵ α, there are x, y L such that x < D < y. We easily see this is equivalent to the following. Lemma Let α be an ordinal number. Let be a total ordering of L. is an η α -ordering of L iff for all subsets A, B of L of cardinality less than ℵ α and such that A < B there is c L with A < c < B. In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set. We use ordinal numbers to simplify some notations. There is no need to have a good background of ordinals in this thesis. However we use some of its properties. Ordinals are an extension of the natural numbers. Like the natural numbers ordinals can be added, multiplied and exponentiated. Also ordinals can be even or odd and are well-ordered. However for most ordinals there are infinitely many other ordinals which are smaller. Also many ordinals do not have a predecessor. We have that the set ℵ 0 of ordinals is well-ordered and of the same cardinality as N. The set ℵ 1 of ordinals is well-ordered and has cardinality equal to the first set greater then the cardinality of ℵ 0. In general if α is an ordinal number then ℵ α is a well-ordered set of ordinals which is of the smallest cardinality greater the cardinality of all ℵ β, where β < α. Definition In this thesis we are only interested in the special cases where α = 0 or α = 1. Let L be a infinite set and a total ordering of L. is called an η 0 -ordering of L iff: (i) for all non-empty finite A, B L, if A < B then there is y L such that A < y < B, (ii) for every non-empty finite subset A L there are x, y L such that x < A < y. Let L be a uncountable set and a total ordering of L. is called an η 1 -ordering of L iff: (i) for all non-empty at most countable sets A, B L, if A < B then there is y L such that A < y < B, (ii) for every set non-empty at most countable set A L there are x, y L such that x < A < y. Example The standard ordering on the rationals is an η 0 -ordering. This immediately implies that on the reals is an η 0 -ordering too. However, is not an η 1 -ordering on the reals. If A = {0} and B = {q Q : q > 0}, then both sets are countable and A < B. But there is no c R such that A < c < B. We now prove a very simple, but powerful, lemma. Lemma Every dense subset of an η 1 -ordering is an η 1 -ordering. Proof. Let L be a set and a η 1 -order on L. Let D L dense in L. Let A and B, both not empty, subsets of D of cardinality less than L, then there is x L such that A < x < B by (i). Applying (i) again, there is y L such that x < y < B. Hence there is an entire interval of L between A and B. Since D is dense in L there is a l L such that A < l < B. Likewise, from (ii), there exists an entire interval greater and an entire interval smaller then A, so there exists l 0, l 1 L such that l 0 < A < l 1. 36