Completeness of Ordered Fields

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by  James Forsythe Hall

Completeness of Ordered Fields
By
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SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
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FOR THE DEGREE OF BACHELOR OF SCIENCE IN MATHEMATICS
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X AT CALIFORNIA POLYTECHNIC STATE UNIVERSITY
r SAN LUIS OBISPO
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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 ba-
sic 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) com-
pleteness 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 equiva-
lency 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 prop-
erties 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 math-
ematics 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 conver-
gent.
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 complete-
ness 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 com-
pleteness, 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 complete-
ness 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.