Questions tagged [ordered-fields]

Ordered fields are fields which have an additional structure, a linear order compatible with the field structure. This tag is for questions regarding ordered fields and their properties, as well proofs related to un-orderability of certain fields.

Filter by
Sorted by
Tagged with
0
votes
1answer
52 views

Show that there are two total orderings of $\textbf{Q}(\sqrt{2})$ under which it is an ordered field.

Let $\textbf{Q}(\sqrt{2})$ be the set of all real numbers of the form $r + s\sqrt{2}$, with $r,s\in\textbf{Q}$. Show that $\textbf{Q}(\sqrt{2})$ is a subfield of $\textbf{R}$. Show that there are two ...
3
votes
0answers
26 views

The discriminant of positive definite binary quadratic forms in partially ordered rings

Let $A$ be a commutative ring, partially ordered by $\le$. Consider the following proposition: Proposition 1. Let $a, b, c$ be in $A$ and assume $a \ge 0$ and $c \ge 0$. If $a n^2 + 2 b n m + c m^2 \...
1
vote
2answers
54 views

What makes the reals a small set?

The following is more of a soft question than a concrete problem, but it haunts me for quite a while now. The reals can be defined as the maximal Archimedean field. They are tiny in comparison to ...
2
votes
2answers
115 views

Dedekind and Cauchy completeness

I know that the following equivalence holds: $$\text{Cauchy-complete ordered Archimedean field}\Leftrightarrow\text{Dedekind-complete ordered field}$$ I would like to know some concrete examples of a ...
2
votes
1answer
28 views

Is the class of reducts of ordered fields axiomatizable?

Let $(F,+,-,*,0,1,\leq)$ be an ordered field. We know that the class of ordered fields is axiomatizable, by definition. Is the $\{+,-,*,0,1\}$ class of reducts of ordered fields axiomatizable? And if ...
0
votes
3answers
36 views

Total ordering of pure imaginary numbers

Real numbers are a totally ordered set, while complex numbers are not. An intuitive explanation can be given by the fact complex numbers can be represented as points of a plane, while real numbers as ...
1
vote
1answer
16 views

Does the complex field with dictionary order have the least-upper-bound property?

If any clarification is necessary, here is a little definition of what "dictionary" order encompasses. In Rudin's book it was stated in the following way. Let $z=a+bi$ and $w=c+di$ where $z,w$ are ...
-1
votes
1answer
68 views

Cauchy completion of transfinite “rationals”

Let the Hessenberg power $\alpha^\beta$ be the supremum of ordinals that are order-isomorphic to some well-order on the set of finite-support functions $\beta \rightarrow \alpha$ that extends the ...
0
votes
0answers
11 views

Can a non-total order define a prepositive cone in a field?

Fact: A total order $\le$ that satisfies the ordered field axioms defines a set $\{x:0\le x\}=P$ that is a prepositive cone. I proved this. I had to invoke trichotomy (as well as other lemmas) to ...
0
votes
1answer
19 views

Proof that the nonnegativy of squares does not follow from the rest of the prepositive cone definition

A prepositive cone $P$ is a subset of a field $F$ defined as exhibiting the following properties: Additive closure: $p+q\in P$; Multiplicative closure: $p\cdot q\in P$; No negative identity: $-1\...
2
votes
0answers
88 views

$b^{x+y}=b^x b^y$ for $x$, $y \in \mathbb{R}$ and $b>1$.

This is from Baby Rudin and it seems to have caused a lost of confusion and controversy among classmates and colleagues, but amidst all of this I found it quite simple and was wondering if I was ...
0
votes
1answer
35 views

Name and definition of the standard ordering on $\mathbb{R}$

Is there any name for the standard ordering on the reals? I typically just see descriptors like "the usual ordering", "the typical ordering", or - well - "the standard ordering". But is there any ...
1
vote
2answers
37 views

Is it true that if $p,q \in K_{>0}$, then $[0,pq) \subseteq [0,p)[0,q)$ in any ordered field?

It seems possible that the following holds: Conjecture. Let $K$ denote an ordered field. If $p,q \in K_{>0}$, then $[0,pq) \subseteq [0,p)[0,q).$ More precisely: If $p,q \in K_{>0}$,...
1
vote
1answer
37 views

How can we show that if $|x| \le 1/n$ for all natural numbers, n, then $x = 0$?

I was thinking about how to define the real number system axiomatically, and can't find anywhere a proof that $$\left[\forall n \in \mathbb{N}\left(|x| \le \frac{1}{n}\right)\right] \Rightarrow [x = 0]...
3
votes
1answer
22 views

Are order-preserving field embeddings unique?

Let $K$ be an ordered field with an embedding into $\mathbb R$, $$f:K\hookrightarrow\mathbb R,$$ where $f$ is order preserving. Is $f$ unique? Follow up from this question of mine (same question ...
1
vote
2answers
18 views

Total ordering on a field defined by a subset

This is an exercise in my analysis class, which I only could finish partially. Let $(F,+,\cdot)$ be a field and let $\mathbb{P} \subset F$ be a set with the following properties: 1) $0 \in \mathbb{...
2
votes
1answer
64 views

Examining the theorem, for all $\epsilon \gt 0$, if $x \le y + \epsilon$, then $x \le y$

For this theorem, I understand the case in which x $\lt y + \epsilon$. From there it follows visually that x could either be equivalent to y itself, or x could also be less than y; in either case, the ...
2
votes
1answer
48 views

Can the Dedekind completion of the hyperreal numbers be embedded in an ordered field?

This answer shows that the Dedekind completion of the set of hyperreal numbers, endowed with the usual definitions of addition and multiplication of Dedekind cuts, is not an ordered field. But my ...
0
votes
0answers
24 views

How to rearrange distance matrix in order of (dis)similarity for coordinates in d-dimensions?

I am trying to self-learn more about distance matrices and their application in different clustering techniques. Suppose I have a distance matrix for N coordinates (...
0
votes
0answers
23 views

Inductive sets in ordered field

In a real analysis book, I read that an ordered field has at least 2 inductive sets and also can have many more. I am not able to understand how "at least 2"? Please clarify.
0
votes
1answer
18 views

Relation between multiplicative and additive identity in an ordered field

In class, we prove that 1>0 by using order axioms of a field. The question is can we conclude that multiplicative identity is always less than the additive identity for any arbitrary ordered field?
1
vote
2answers
54 views

To prove (2,2) is an interval in R

The definition of an interval in an ordered field is a set that always contains the entire closed interval between any two of its elements. How do we prove (2,2) is an interval which is an empty set ...
0
votes
1answer
49 views

A Sequence which is co-Cauchy with a Positive Cauchy Sequence is Positive

Indeed, this question arises when one wants to prove the well-definedness of an order while constructing $\mathbb{R}$ from equivalence classes of Cauchy sequences in $\mathbb{Q}$. Definition 1. A ...
0
votes
2answers
51 views

show $1/x$ does not equal $ 0$ [closed]

Suppose that $x\neq 0$, show that $\frac{1}{x}\neq 0$ This seems like it should be simple, but I don't even know how to start to prove this.
1
vote
1answer
34 views

A property of $\gamma_1$ extensions of real closed fields.

For each real closed ordered field $\mathbb{F}$, $\sum_\omega({}^m\mathbb{F},S)$ denotes the set of all definable subsets in ${}^m\mathbb{F}$ with parameters in S. We consider $\sum_\omega({}^m\mathbb{...
1
vote
1answer
37 views

Show that extension $\mathbb{R}(x) \supseteq \mathbb{R}$ is a real extension (ordered extension)

Problem: Denote $\mathbb{R}(x)$ be the quotient field of polynomial ring in single variable $\mathbb{R}[x]$. Show that extension $\mathbb{R}(x) \supseteq \mathbb{R}$ is a real extension (ordered ...
0
votes
1answer
43 views

Show that the real field $\mathbb{R}$ has a unique ordering and indicates that ordering

Problem: Show that the real field $\mathbb{R}$ has a unique ordering and indicates that ordering. My question: We knew that $\le$ is an ordering on $\mathbb{R}$. Do we need to prove that $\le$ is an ...
3
votes
2answers
61 views

Maximal algebraic ordered field extensions of $\mathbb{Q}$

The real algebraic numbers form a maximal algebraic ordered field extension of $\mathbb{Q}$ in the sense that they are an algebraic ordered field extension of $\mathbb{Q}$, and no other ordered field ...
1
vote
1answer
55 views

Archimedean Property and limit of $q^n$ for $0<q<1$

How could I show, that if $\lim_{n\rightarrow\infty}q^n=0$ for $0<q<1$ in an ordered field (not $\mathbb{R}$), that the ordered field is archimedean, meaning for every $x,y$ in this field with $...
2
votes
1answer
39 views

Can a real-closed field of uncountable cofinality have a countable gap?

Let $\mathbb{F}$ be a real-closed field with $\mathrm{cof}(\mathbb{F})\geq\omega_1$. For subsets $A$ and $B$ of $\mathbb{F}$, we say that $\langle A, B \rangle$ is a pregap if $\forall a \in A \ \...
0
votes
2answers
53 views

Proof that if $\lim_{n \to \infty } \frac{a(n+1)}{a(n)} =0$ for $a:\mathbb{N}\mapsto K, a(n)>0 \forall n\in \mathbb{N}$, then $\lim(a)=0$

I want to prove: Is $K$ an ordered field and $a:\mathbb{N}\rightarrow K$ with $a(n)>0\;\forall n\in \mathbb{N}$. If $\lim_{n \to \infty} \frac{a(n+1)}{a(n)} =0$, then $\lim a=0$. Please let me ...
2
votes
1answer
65 views

Understanding why complicated argument is necessary when looking for index 2 subgroups

I have question concerning the proof of Theorem 1 in On ordered skew fields. The author uses Zorns lemma to construct a maximal subgroup $P \subset K^*$ with $-1 \notin P$ and $S \subseteq P$. We know ...
0
votes
0answers
26 views

Solve the problem with numbers from 1 to 101 written in a random order

Numbers from 1 to 101 are written in a row in a random order forming a list. Prove that you can remove 90 numbers from this list such that remaining numbers will be sorted in the ascending or ...
2
votes
2answers
63 views

A definition of ordered field

Definition: An ordered field is field $F$ equipped with a linear order $<$ such that $1 > 0$ For any $a,b \in F$, $$ a,b > 0 \quad \Longrightarrow \quad a+b,\;ab > 0.$$ I like this ...
0
votes
0answers
41 views

Is $\mathbb{Q}$ the smallest ordered field up to isomorphism?

Pretty simple question. Does there exist a ordered field smaller than (i.e. is a strict subset of) $\mathbb{Q}$? It seems like we can't go any smaller than $\mathbb{Q}$. Is this true? Why?
3
votes
1answer
34 views

No single axiom stating non-Archimedeanity [duplicate]

An ordered field $K$ with the ordering $<$ is Archimedean if for any $x \in K$ there exists $n \in \mathbb{N}$ (where $\mathbb{N}$ is the copy of the natural numbers in $K$) such that $|x|<n$. ...
0
votes
1answer
23 views

Proof of $c > a - 2b$ if $|c - a| < b$

Let $(K,+,\cdot, P)$ be a totally ordered field. How can one prove that for $a,b,c \in K$ if $|c - a| < b$ it holds that $c > a - 2b$? I tried it out with numbers and couldn't find a ...
0
votes
1answer
17 views

Proof of $ca^2 < cb^2$ in an ordered field

Let $(K,+,\cdot, P)$ be a totally ordered field. How can one prove that for $a,b,c \in K$ it holds that if $a < b$ and $c > 0$ then $ca^2 < cb^2$? I know that if $a < b \wedge c > 0$ ...
0
votes
1answer
30 views

How can I prove this statement without using reduction to absurdity?

$\forall a,b\in\mathbb R[\forall c\in \mathbb R(c>a\implies c>b)\implies a\ge b]$
0
votes
0answers
31 views

First-order properties of Euclidean fields (instead of real closed fields)

Let $\mathbb{F}$ be an ordered field. $\mathbb{F}$ is called Euclidean if $\forall x>0 \in \mathbb{F}\ \exists y \in \mathbb{F} : y^2=x$, i.e. a square root exists. $\mathbb{F}$ is called real ...
1
vote
1answer
51 views

What does “$\mathbb{R}$-complete” mean in Woodin's paper?

I'm reading Woodin's paper "A Discontinuous Homomorphism from C(X) without CH". In this paper, Woodin defined "$\mathbb{R}$-complete" as the following: DEFINITION 1. Suppose that $\mathbb{H} \subset ...
-1
votes
1answer
46 views

use the ordered field $\mathbb{R}$. For $|x−y|<0.01$ and $x,y\in(0,2),$ show that $|x^2−y^2|<0.04$ only use properties of absoulte values [closed]

I Don't know how to start this question can someone please help me? Use the ordered field $\mathbb{R}$. For $|x−y|<0.01$ and $x,y\in(0,2),$ show that $|x^2−y^2|<0.04$
1
vote
1answer
49 views

Equivalence relation on ordered field

Let p be a prime number. Define an equivalence relation ∼ on Z as: n ∼ m if n−m is divisible by p. For n ∈ Z, let [n] be the equivalence class of n with respect to this equivalencerelation. LetZp ={[n]...
0
votes
1answer
28 views

“Simple” ordered fields proof: show that $0<y^{-1}<1$

Given that $x>0$ and $y=x+1$ show that $0<y^{-1}<1$ specifying what proprieties of the Ordered Field you are using. I really struggle to understand whether I have demonstrated the theorem ...
2
votes
1answer
80 views

Galois Group of Ordered Field containing Square Roots

Given a ordered field $K$ such that every positive element $0<x$ has a root in $K$ we can show that any endomorphism $f:K\rightarrow K$ preserves the ordering of $K$ and turns out to be $\text{id}...
1
vote
3answers
40 views

Is it useful that $k(X)$ is an ordered field if $k$ either is?

I have recently found that if $k$ is an ordered field, $k(X)$ is also an ordered field. Proof below, in the proof that order is defined. My question: I have been never told that a field of ...
1
vote
2answers
108 views

Prove that there exists no such total order

Prove that there does not exist a total order $\leq$ on $\mathbb{C}$ such that (i) for all $x,y,z \in \mathbb{C}$, if $x\leq y,$ then $x+z\leq y + z$; (ii) for all $x, y \in \mathbb{C}$, if $x \geq ...
1
vote
2answers
70 views

The Order of sets like $\mathbb Q$ and Justification of the Number Line

Rudin, in Principles of Mathematical Analysis, defines an ordered set $S$ as a set with a relation such that (i) If $x\in S$ the one and only one of the statements $$x<y,\,\,\,\,x=y\,\,\,\,\,y&...
3
votes
1answer
64 views

Find a transfinite monotone subsequence.

Let $(\mathbb{K},0,1,+,\times,\leq)$ be an ordered field. Let $\delta$ be its cofinality, i.e. the length of the smallest sequence of $\mathbb{K}$ that is cofinal with it. $\delta$ is a regular ...
2
votes
1answer
90 views

Axioms of order in geometry and ordered fields

I am considering axioms of incidence and axioms of order for plane geometry by Hilbert: I1: Two points determine the unique line. I2: Each line contains (at least) two points. I3: There are three ...

1
2 3 4 5
7