Completeness of Ordered Fields By 1 James Forsythe Hall 1 0 2 n a J 9 2 ] O L . h t a m [ 1 v 2 5 6 5 . 1 0 1 1 SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS : v FOR THE DEGREE OF BACHELOR OF SCIENCE IN MATHEMATICS i X AT CALIFORNIA POLYTECHNIC STATE UNIVERSITY r SAN LUIS OBISPO a DECEMBER 2010 (cid:13)c2010JamesHall Keywords Totally ordered fields · Dedekind fields · Complete fields · Infinitesimals · Non- standard analysis · Nonstandard extension · Archimedean fields · Saturation principle · Power-series fields · Hahn field · Robinson asymptotic numbers Mathematics Subject Classification (2010)Primary: 12J15; Secondary: 03H05·06A05 · 12J25 · 26E35 ThisisaseniorprojectdoneunderthesupervisionofTodorD.Todorov Introduction In most textbooks, the set of real numbers R is commonly taken to be a totally ordered Dedekind complete field. Following from this definition, one can then establish the basic properties of R such as the Bolzano-Weierstrass property, the Monotone Convergence property, the Cantor completeness of R (Definition 3.1), and the sequential (Cauchy) completeness of R. The main goal of this project is to establish the equivalence of the preceding properties, in the setting of a totally ordered Archimedean field (Theorem 3.11), along with a less well known algebraic form of completeness (Hilbert completeness, Definition 3.1) and a property from non-standard analysis which roughly states that every finite element from the non-standard extension of the field is “infinitely close” to some element from the field (Leibniz completeness, Definition 3.1). (The phrase infinitely close may be off-putting to some as it is sometimes associated with mathematics lacking in rigour, but in section §2 we properly define this terminology and in section §8 and §9 we provide several examples of fields with non-trivial infinitesimal elements.) As is usual in mathematics, we continued our research past that of Archimedean fields to determine how removing the assumption that the field is Archimedean affects the equivalency of the properties listed in Theorem 3.11. What we found is that all but three of those properties are false in non-Archimedean fields: Hilbert completeness, Cantor completeness and sequential completeness. Furthermore, we found that the last two of these three properties are no longer equivalent; rather, the latter is a necessary, but not sufficient, condition for the former (see Theorem 6.6 and Example 9.3). One application of Theorem 3.11 is given in section §4, where we establish eight different equivalent collections of axioms, each of which can be used as axiomatic definitions of the set of real numbers. Another application is an alternative perspective of classical mathematics that results from the equivalency of Dedekind completeness (Definition 3.1) and the non-standard property outlined in the first paragraph above (this property is defined in Definition 3.1). 1 Orderable Fields In this section we recall the main definitions and properties of totally ordered fields. For more details, we refer to Lang [11] and van der Waerden [32]. 1.1 Definition (Orderable Field). A field K is orderable if there exists a non-empty K ⊂ K + such that 1. 0 6∈ K + 1 1. Orderable Fields 2 2. (∀x,y ∈ K )(x+y ∈ K and xy ∈ K ) + + + 3. (∀x ∈ K\{0})(x ∈ K or −x ∈ K ) + + Provided that K is orderable, we can fix a set K that satisfies the properties given above + and generate a strict order relation on K by x < y iff y − x ∈ K . Further, we can K+ + define a total ordering (i.e. reflexive, anti-symmetric, transitive, and total) on K by x ≤ y K+ iff x < y or x = y. K+ 1.2 Definition (Totally Ordered Field). Let K be an orderable field and K ⊂ K. Then + using the relation ≤ generated by K , as noted above, we define (K,≤ ) to be a totally K+ + K+ ordered field. As well, with this ordering mind, we define the absolute value of x ∈ K as |x| =: max(−x,x). For simplicity, when K is clear fromthe context we shall use the more standard notation + x ≤ y and x < y in place of x ≤ y and x < y, respectively. As well, we shall refer to K+ K+ K as being a totally ordered field – or the less precise, ordered field – rather than the more cumbersome (K,≤ ). K+ 1.3 Lemma (Triangle Inequality). Let K be an ordered field and a,b ∈ K, then |a+b| ≤ |a|+|b|. Proof. As −|a|−|b| ≤ a+b ≤ |a|+|b|, we have |a+b| ≤ |a|+|b| because a+b ≤ |a|+|b| and −(a+b) ≤ |a|+|b|. 1.4 Definition. Let K be a totally ordered field and A ⊂ K. We denote the set of upper bounds of A by UB(A). More formally, UB(A) =: {x ∈ K : (∀a ∈ A)(a ≤ x)} 1.5 Definition (Ordered Field Homomorphism). Let K andF beordered fields and φ : K → F be a field homomorphism. Then φ is called an ordered field homomorphism if it preserves the order; that is, x ≤ y implies φ(x) ≤ φ(y) for all x,y ∈ K. Definitions of ordered field ismorphisms and embeddings follow similarly. 1.6 Remark. Let K be an ordered field. Then we define the ordered field embedding σ : Q → K by σ(0) =: 0, σ(n) =: n· 1, σ(−n) =: −σ(n) and σ(m) =: σ(m) for n ∈ N and k σ(k) m,k ∈ Z. We say that σ is the canonical embedding of Q into K. 1.7 Definition (Formally Real). A field K is formally real if, for every n ∈ N, the equation n x2 = 0 k k=0 X 1. Orderable Fields 3 has only the trivial solution (that is, x = 0 for each k) in K. k 1.8 Theorem. A field K is orderable iff K is formally real. Further discussion, as well as the proof, of the preceding theorem can be found in van der Waerden [32] (chapter 11). 1.9 Example (Non-Orderable Field). 1. The field of complex numbers C is not orderable. Indeed, suppose there exists a subset C ⊂ C that satisfies the properties above. Thus i ∈ C or −i ∈ C . However, either + + + case implies that −1 = (±i)2 ∈ C and 1 = (−1)(−1) ∈ C . Thus 0 = 1−1 ∈ C , a + + + contradiction. Therefore, C is non-orderable. 2. The p-adic numbers Q are also non-orderable for similar reasons (see Ribenboim [19] p p.144-145 and Todorov & Vernaeve [26]) 1.10 Definition (Real Closed Field). Let K be a field. We say that K is a real closed field if it satisfies the following. 1. K is formally real (or orderable). 2. (∀a ∈ K)(∃x ∈ K)(a = x2 or a = −x2). 3. (∀P ∈ K[t])(deg(P) is odd ⇒ (∃x ∈ K)(P(x) = 0)). 1.11 Theorem. Let K be a real closed totally ordered field and x ∈ K. Then x > 0 iff x = y2 for some y ∈ K. Thus every real closed field is ordered in a unique way. Proof. Suppose x > 0, then there exists y ∈ K such that x = y2 by part 2 of Definition 1.10. Conversely, suppose x = y2 for some y ∈ K. By the definition of K , we have y2 ∈ K + + for all y ∈ K. Thus x > 0. 1.12 Remark. If the field K is real closed, then we shall always assume that K is ordered by the unique ordering given above. 1.13 Lemma. Let K be an ordered field and a ∈ K be fixed. The scaled identity function a · id(x) =: ax is uniformly continuous in the order topology on K. Consequently, every polynomial in K is continuous. Proof. Given ǫ ∈ K , let δ = ǫ . Indeed, (∀x,y ∈ K)(|x−y| < δ ⇒ |ax−ay| = |a||x−y| < + |a| |a|δ = ǫ). 1. Orderable Fields 4 1.14 Lemma (Intermediate Value Theorem). Let K be an ordered field. As well, let f : H → K be a function that is continuous in the order topology on K and [a,b] ⊂ H. If K is Dedekind complete (in the sense of sup), then, for any u ∈ K such that f(a) ≤ u ≤ f(b) or f(b) ≤ u ≤ f(a), there exists a c ∈ [a,b] such that f(c) = u. Proof. We will only show the case when f(a) ≤ f(b), the other should follow similarly. Let S =: {x ∈ [a,b] : f(x) ≤ u}. We observe that S is non-empty as a ∈ S and that S is bounded above by b; thus, c =: sup(S) exists by assumption. As well, we observe that c ∈ [a,b] because c ≤ b. To show that f(c) = u we first observe that, as f is continuous, we can find δ ∈ K such that (∀x ∈ K)(|x−c| < δ ⇒ |f(c)−f(x)| < |f(c)−u|). + If f(c) > u, then, from our observation, it follows that f(x) > f(c)−(f(c)−u) = u for all x ∈ (c−δ,c+δ). Thus c−δ ∈ UB(S), which contradicts the minimality of c. Similarly, ifu > f(c),then, fromourobservationitfollowsthatu = f(c)+(u−f(c) > f(x) for all x ∈ (c−δ,c+δ), which contradicts c being an upper bound. Therefore f(c) = u as K is totally ordered. 1.15 Remark. When dealing with polynomials, it follows from the Artin-Schrier Theorem that Dedekind completeness is not necessary toproduce the results of the Intermediate Value Theorem. For a general reference, see Lang [11], Chapter XI. 1.16 Theorem. Let K be a totally ordered field which is also Dedekind complete. Then K is real closed. Proof. First observe that K is formally real because it is orderable. Now let a ∈ K andS =: {x ∈ K : x2 < a}. Observe that0 ∈ S andthatm =: max{1,a} + is an upper bound of S. Indeed, when a ≤ 1, we have x2 < 1, which implies x < 1 for all x ∈ S. On the other hand, when 1 < a, we have x2 < a < a2; thus, x < a for all x ∈ S. From this observation, it follows that s =: supS exists. We intend to show that s2 = a. Case (s2 < a): Let h =: 1 min{ a−s2 ,1}. From this definition, it follows that 2 (s+1)2 a−s2 2h ≤ and 2h ≤ 1 (1) (s+1)2 We wish to show that (s+h)2 < a. From 0 < h ≤ 1 we have h < s2 + 1, which implies that h + 2s < (s + 1)2 and 2 h(h+2s)+s2 < h(s+1)2 +s2. Thus, we have (s+h)2 < h(s+1)2 +s2. By (1), we know that h(s+1)2 < 2h(s+1)2 ≤ a−s2. Thus, we have (s+h)2 < h(s+1)2+s2 < a. Therefore (s+h) ∈ S which contradicts s being an upper bound. 2. Infinitesimals in Ordered Fields 5 Case (s2 > a): Let h =: s2−a . First we observe that, from the definition of h, s2 − a = 2(s+1)2 2h(s+ 1)2 > h(s+ 1)2. We intend to show that (s−h)2 > a. Indeed, we obviously have s2 + 1 > −h which implies that (s + 1)2 > 2s − h and h(s + 1)2 > h(2s − h). Thus s2 − h(s + 1)2 < s2 − h(2s − h) = (s − h)2 and by our observation, we find that a < s2 −h(s+1)2 < (s−h)2. Therefore (s−h) is an upper bound of S, which contradicts the minimality of s. Finally, toshowthateveryodddegreepolynomialP(x) ∈ K[x]hasaroot,weobservethat lim P(x) = −lim P(x). Combining this result with Lemma 1.13 and Lemma 1.14, x→−∞ x→∞ we find that ∃c ∈ K such that P(c) = 0. 2 Infinitesimals in Ordered Fields In this section we recall the definitions of infinitely small (infinitesimal), finite and infinitely large elements in totally ordered fields and study their basic properties. As well, we present a characterization of Archimedean fields in the languague of infinitesimals and infinitely large elements. 2.1 Definition (Archimedean Property). A totally ordered field (ring) K is Archimedean if for every x ∈ K, there exists n ∈ N such that |x| < n. If K is Archimedean, we also may refer to K(i) as Archimedean. If K is not Archimedean, then we refer to K as non-Archimedean. FortherestofthesectionwediscussthepropertiesofArchimedean andnon-Archimedean fields through the characteristics of infinitesimals. 2.2 Definition. Let K be a totally ordered field. We define 1. I(K) =: {x ∈ K : (∀n ∈ N)(|x| < 1)} n 2. F(K) =: {x ∈ K : (∃n ∈ N)(|x| ≤ n)} 3. L(K) =: {x ∈ K : (∀n ∈ N)(n < |x|)} The elements in I(K),F(K), and L(K) are referred to as infinitesimal (infinitely small), finite and infinitely large, respectively. We sometimes write x ≈ 0 if x ∈ I(K) and x ≈ y if x−y ≈ 0, in which case we say that x is infinitesimally close to y. 2.3 Proposition. For a totally ordered field K, we have the following properties for the sets given above. (i) I(K) ⊂ F(K). 2. Infinitesimals in Ordered Fields 6 (ii) K = F(K)∪L(K). (iii) F(K)∩L(K) = ∅. (iv) If x ∈ K\{0} then x ∈ I(K) iff 1 ∈ L(K). x Proof. (i) Let α ∈ I(K), then α < 1, therefore α ∈ F(K). (ii) Suppose x ∈ K, then either (∃n ∈ N)(|x| < n) or (∀n ∈ N)(n ≤ |x|). Thus x ∈ F(K) or x ∈ L(K). The other direction follows from the definition. (iii) Suppose x ∈ F(K) ∩ L(K), then ∃n ∈ N such that |x| < n, but, we also have (∀m ∈ N)(m < |x|); thus n < |x| < n, a contradiction. (iv) Suppose x ∈ K\{0}. Then x ∈ L(K) iff (∀n ∈ N)(n < |x|) iff (∀n ∈ N)(|1| < 1) x n iff 1 ∈ I(K) x 2.4 Proposition (Characterizations). Let K be a totally ordered field. Then the following are equivalent: (i) K is Archimedean. (ii) L(K) = ∅. (iii) I(K) = {0}. (iv) F(K) = K. Proof. (i) ⇒ (ii) Follows from the definition of Archimedean field. (ii) ⇒ (iii) Suppose dα ∈ I(K) such that dα 6= 0. As Kis a field, dα−1 exists. Thus dα < 1, n for all n ∈ N, which gives 1 < 1dα−1 for all n ∈ N. Therefore n < dα−1 for all n ∈ N, n which means dα ∈ L(K). (iii) ⇒ (iv) Note that we clearly have F(K) ⊆ K. Suppose, to the contrary, there exists α ∈ K\F(K). Then, by definition, |α| > n for all n ∈ N; hence 1 > 1 for all n ∈ N n |α| because K is a field. Thus 1 ∈ I(K) so that 1 = 0, a contradiction. |α| |α| (iv) ⇒ (i) By definition of F(K), we know that for every α ∈ K = F(K) there exists a n ∈ N such that |α| < n; hence K is Archimedean. 2.5 Lemma. Let K be a totally ordered field. Then 3. Completeness of an Archimedean Field 7 (i) F(K) is an Archimedean ring. (ii) I(K) is a maximal ideal of F(K). Moreover, I(K) is a convex ideal in the sense that a ∈ F(K) and |a| ≤ |b| ∈ I(K) implies a ∈ I(K). Consequently F(K)/I(K) is a totally ordered Archimedean field. Proof. (i) The fact that F(K) is Archimedean follows directly from its definition. Observe that |−1| ≤ 1, therefore −1 ∈ F(K). Suppose that a,b,c ∈ F(K), then |a| ≤ n, |b| ≤ m and |c| ≤ k for some n,m,k ∈ N. Thus |ab+c| ≤ |a||b|+|c| ≤ nm+k ∈ N by Lemma 1.3, which implies ab+c ∈ F(K). (ii) Let x,y ∈ I(K). Then, for any n ∈ N, we have |x+y| ≤ |x|+|y| < 1 + 1 = 1; 2n 2n n thus x+y ∈ I(K). Now suppose a ∈ I(K) and b ∈ F(K). Then |b| ≤ n for some n ∈ N. As |a| < 1 nm for all m ∈ N, we have |ab| ≤ n = 1 for all m ∈ N. Hence, ab ∈ I(K). nm m Suppose there exists an ideal R ⊆ F(K) that properly contains I(K) and let k ∈ R \ I(K). Then 1 ≤ |k| for some n ∈ N, hence n ≥ 1 ∈ K which implies n |k| 1 ∈ F(K) and 1 = k ∈ R. Therefore R = F(K). k k Finally, let b ∈ I(K). Suppose a ∈ F(K) such that |a| < |b|. Then |a| < |b| < 1 n for all n ∈ N. Therefore a ∈ I(K). 2.6 Remark. Archimedean rings (which are not fields) might have non-zero infinitesimals. For example, F(K) is always an Archimedean ring, but it has non-zero infinitesimals when K is a non-Archimedean field. 2.7 Example (Archimedean Fields). The fields R,Q,C are all Archimedean fields. For examples of non-Archimedean fields, we refer the reader to § 8 and § 9. 3 Completeness of an Archimedean Field In what follows, convergence is meant in reference to the order topology on K. As well, the reader should recall that there is a natural embedding of the rationals (and, thus the natural numbers) into any totally ordered field (see remark 1.6). In what follows, we provide several definitions of rather well known forms of completeness that will be used throughout the rest of this section. 3. Completeness of an Archimedean Field 8 3.1 Definition (Completeness). Let K be a totally ordered field. 1. Let κ be an uncountable cardinal. Then K is Cantor κ-complete if every family {[a ,b ]} of fewer than κ closed bounded intervals in K with the finite intersection γ γ γ∈Γ property (F.I.P.) has a non-empty intersection, [a ,b ] 6= ∅. If K is Cantor ℵ - γ∈Γ γ γ 1 complete, where ℵ = ℵ+ (the successor of ℵ = card(N)), then we say that K is 1 0 0 T Cantor complete. The latter means that every nested sequence of bounded closed intervals in K has a non-empty intersection. 2. Let ∗K be a non-standard extension of K (see either Lindstrøm [13] or Davis [4]) and let F(∗K) and I(∗K) be the sets of finite and infinitesimal elements in ∗K, respectively (see Definition 2.2). Then we say that K is Leibniz complete if for every α ∈ F(∗K), there exists unique L ∈ K and dx ∈ I(∗K) such that α = L+dx; we will sometimes denote this by F(∗K) = K⊕I(∗K) which is equivalent to saying F(∗K)/I(∗K) = K. 3. K is Dedekind complete if every non-empty subset of K that is bounded from above has a supremum. 4. K is sequentially complete if every fundamental (Cauchy) sequence in K converges. Recall that a sequence {a } in a totally ordered field K (not necessarily Archimedean) n iscalledfundamental ifforallǫ ∈ K , thereexistsanN ∈ Nsuchthatforalln,m ∈ N, + n,m ≥ N implies that |a −a | < ǫ. n m 5. We say that K is Bolzano-Weierstrass complete if every bounded sequence has a convergent subsequence. 6. We say that K is Bolzano complete if every bounded infinite set has a cluster point. 7. We say that K is monotone complete if every bounded monotonic sequence is convergent. 8. Suppose that K is Archimedean. Then K is Hilbert complete if K has no proper totally ordered Archimedean field extensions. 3.2 Remark (Completeness of the Reals in History). 1. Leibniz completeness (number 2 in Definition 3.1 above) appears in the early Leibniz- Euler Infinitesimal Calculus as the statement that “every finite number is infinitesmi- ally close to a unique usual quantity.” Here the “usual quantities” are what we now refer to as the real numbers and can be identified with K in the definition above. This form of completeness was more or less, always treated as an obvious fact; what 3. Completeness of an Archimedean Field 9 was not obvious, and a possible reason for the demise of the infinitesimals, was the validity of what has come to be known as the Leibniz Principle (see H. J. Keisler [8] p. 42 and Stroyan & Luxemburg [23] p. 22), which roughly states that there is a non- Archimedean field extension ∗K of K such that every function f on K has an extension ∗f to ∗K that “preserves all the properties of K.” For example, (x+y)2 = x2+2xy+y2 and ∗sin(x+y) = ∗sin(x)∗cos(y)+∗sin(y)∗cos(x) hold in ∗K because the analogous statements hold in K (note that ∗sin is usually written as sin for convenience). All attempts to construct a field with such properties failed until the 1960’s when A. Robinson developed the theory of non-standard analysis along with the Transfer Prin- ciple, which is analogous to the Leibniz Principle, and proved that every field K has a non-standard extension ∗K. For a detailed exposition, we refer to Lindstrøm [13], Davis [4] and Keisler [8],[9]. 2. Dedekind completeness was introduce by Dedekind (independently from many others, see O’Connor [17]) at the end of the 19th century. From the point of view of modern mathematics, Dedekind proved the consistency of the axioms of the real numbers by constructing an example from Dedekind cuts. 3. Sequential completeness, listed as number 4 above, is a well known form of completeness of metric spaces, but it has also been used in constructions of the real numbers: Cantor’s construction using Cauchy sequences (see O’Connor [17]), an example of which can be found in Hewitt & Stromberg [6]. 4. Cantor completeness (also known as the “nested interval property”), monotone completeness, Bolzano-Weierstrass completeness, and Bolzano completeness typically ap- pear in real analysis as “theorems” or “important principles” rather than as forms of completeness; however, in non-standard analysis, Cantor completeness takes a much more important role along with the concept of algebraic saturation which is defined in Definition 6.2. 5. Hilbert completeness, listed as number 8 above, is a less well-known form of completeness that was originally introduced by Hilbert in 1900 with his axiomatic definition of the real numbers (see Hilbert [7] and O’Connor [17]). 3.3 Theorem. There exists a Dedekind complete field. Proof. For the proof of this we refer to either the construction by means of Dedekind cuts in Rudin [22] or the construction using equivalence classes of Cauchy sequences in Hewitt & Stromberg [6], or to § 5 of this text where we present a construction using non-standard analysis.

Completeness of Ordered Fields

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