# Questions tagged [ordered-fields]

The ordered-fields tag has no usage guidance.

38
questions

**4**

votes

**1**answer

251 views

### On a completeness property of hyperreals

Let $\mathbb{R}_*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of this question and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}_*$: $(\omega,\...

**4**

votes

**0**answers

74 views

### Witt ring of a field with Pythagoras number $2$

I am currently looking at a few simple properties of the Witt ring of a field $K$ (by which I mean the ring of Witt classes of quadratic forms, not the ring of Witt vectors), which are clearly true ...

**7**

votes

**0**answers

96 views

### Reduced power of an ordered field

Suppose $K$ is an ordered field, $X$ is any set, and $F$ is a filter over $X$. Let $G$ be the reduced power of $K$ by $F$. That is, we take all functions from $X$ to $K$, then take equivalence ...

**24**

votes

**6**answers

2k views

### Is this theory the complete theory of the real ordered field?

We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider ...

**5**

votes

**1**answer

277 views

### Is there a complete characterization of ordered fields without definable proper subfields?

$\mathbb{Q}$ has no proper subfields. As a result, all ordered fields elementarily equivalent to $\mathbb{Q}$ have no proper subfields which are first-order definable without parameters. And by the ...

**2**

votes

**0**answers

89 views

### Request for bibliographic information

Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding:
...

**11**

votes

**2**answers

456 views

### Decidability of a first-order theory of hyperreals

The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.
If we add a unary ...

**0**

votes

**0**answers

60 views

### Tensor product of preordered rings

All rings in this post are commutative, unital, and contain $\frac{1}{2}$.
To study "real" properties of a ring $R$, one is often interested in the orderings which exist on fraction fields of ...

**17**

votes

**2**answers

540 views

### Are radicals dense in the real closure of an ordered field?

Let $F$ be an ordered field and let $R$ denote its real closure.
It is well-known that $F$ is cofinal in $R$, but not necessarily dense.
For example, consider $F=\mathbb{R}(\omega)$ with the order ...

**57**

votes

**8**answers

7k views

### Cauchy reals and Dedekind reals satisfy "the same mathematical theorems"

The succinct question
The conjecture of Birch and Swinnerton-Dyer (to take a random example) mentions L-functions and hence the complex numbers and hence the real numbers (because the complexes are ...

**11**

votes

**3**answers

702 views

### Are archimedean subextensions of ordered fields dense?

Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$.
Is it true that if $E$ is $F$-...

**7**

votes

**2**answers

645 views

### How do fractional tensor products work?

[I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...

**3**

votes

**1**answer

148 views

### 'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group

What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...

**7**

votes

**2**answers

311 views

### Formally real fields with unique non-Archimedean ordering

My question is rather simple. Do there exist a formally real field that admits a unique ordering (so sums of squares are the positive elements) and such that this ordering is not archimedean?
Oh, I ...

**2**

votes

**0**answers

97 views

### Atomic integer parts

Let $R$ be an ordered ring (in particular, the order is linear and $R$ is a domain). Let $| \ \ |: x \mapsto \max(x,-x)$ denote its absolute value.
For $(x,y) \in R \times R^{\neq 0}$ say that an ...

**4**

votes

**0**answers

238 views

### Do all fields with internal absolute values arise as ordered fields or like $\mathbb{C}$ from them?

$\def\abs#1{\lvert#1\rvert}
\def\Im{\operatorname{Im}}
\def\Re{\operatorname{Re}}$
(Crossposted from math.stackexchange.com after 5 days with no correct answer.)
Let $\langle F,+,\cdot\rangle$ be ...

**1**

vote

**1**answer

154 views

### Archimedean completeness of some fields

I need a reference (different from Hahn's 1907 paper) for the following result.
Theorem: If $G$ is a totally ordered abelian group, then the field $\mathbb{R}((G))$ is archimedean complete.
$\...

**1**

vote

**1**answer

76 views

### Bound for annihilating polynomials

Let $F$ be an ordered field, let $L$ be the real closure of $F$.
Let $R \in L$ be strictly positive. Can one find a bound $M \geq 0$ and for each $x \in ]-R;R[_L$, an element $x' \in [x-1;x+1]_L$ ...

**5**

votes

**1**answer

153 views

### Is there an exponential map on (Hahn) ordered fields?

If $F$ is an ordered field and $G$ is an ordered abelian group, one can define the Hahn product $F \boxtimes G$ to be the set of formal Laurent series with coefficients in $F$ and exponents in $G$. It ...

**4**

votes

**1**answer

242 views

### Completing class-sized Fields

Let's say that an ordered Field is a class (proper or not) which satisfies the axioms of ordered fields. We work in NBG set theory with global choice.
Let's say that an ordered Field is real closed ...

**4**

votes

**2**answers

345 views

### Which ordinals can be embedded into an ordered field?

Let $F$ be an ordered field.
What is the least ordinal $\alpha$ such that there is no order-embedding of $\alpha$ into any bounded interval of $F$?

**3**

votes

**1**answer

157 views

### Least ordinal not embedded in a total order

If $(E,<)$ is a linear order, let $s(E,<)$ denote the least ordinal which doesn't embed in $(E,<)$.
I am trying to prove the following:
If $(M,+,.,0,1)$ is a model of open induction, (or ...

**5**

votes

**1**answer

164 views

### Ordinals which embed in surreal subfields

If $k$ is an ordered field, the least ordinal $s(k)$ which doesn't embed in $(k,<)$ is regular. This is because every interval of an ordered field embeds in every infinite interval so given a ...

**2**

votes

**1**answer

207 views

### Cauchy completeness of the real closure

Let $k$ be an ordered field of cofinality $cf(k)$ whose Cauchy $cf(k)$-sequences are convergent.$^{(1)}$
Let $\mathcal{R}(k)$ be its real closure.
As an algebraic extension of $k$, it has the same ...

**20**

votes

**1**answer

1k views

### Differential Topology over $\mathbb{Q}$

I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it:
Are every two rational manifolds of the ...

**23**

votes

**1**answer

643 views

### Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...

**8**

votes

**2**answers

534 views

### Possible cardinality and weight of an ordered field

Is it true (in ZFC) that for any regular infinite cardinal $\kappa$ there exists an ordered field of weight $\kappa$ and cardinality $2^\kappa$ (or at least $>\kappa$)?
The field of real numbers ...

**31**

votes

**1**answer

3k views

### What did Rolle prove when he proved Rolle's theorem?

Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...

**10**

votes

**3**answers

1k views

### Does this construction yield the surreal numbers?

There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals.
First given such a field one may consider rational functions over that field ...

**0**

votes

**1**answer

102 views

### Ways to order an algebraic extension

In following post, I describe the "classical" example of $\mathbb{Q}(\sqrt{2})$ that can be ordered in two distinct ways.
More generally, if $(k,P)$ is an ordered field, $R$ a real closure of $(k,P)$ ...

**5**

votes

**1**answer

347 views

### Isomorphism of real closed fields

Given two real closed fields $R_1$ and $R_2$ such that both have cardinality continuum, archimedean, but not necessarily complete. Assume further that they are back and forth equivalent (in the ...

**35**

votes

**6**answers

2k views

### On the universal property of the completion of an ordered field

I have been trying to write up some notes on completion of ordered fields, ideally in the general case (i.e., not just completing $\mathbb{Q}$ to get $\mathbb{R}$ but considering the completion via ...

**9**

votes

**1**answer

1k views

### What fields can be used for an inner product space?

Math people:
The title is the question: What fields can be used for an inner product space?
This question has been discussed in Math Stack Exchange with no definitive resolution. A similar question ...

**5**

votes

**0**answers

687 views

### two versions of the nested interval property

There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...

**10**

votes

**2**answers

974 views

### Does Rolle's Theorem imply Dedekind completeness?

I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...

**10**

votes

**0**answers

303 views

### Does antidifferentiability of continuous functions imply Dedekind completeness?

Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...

**12**

votes

**5**answers

2k views

### analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...

**5**

votes

**5**answers

3k views

### What are examples of ordered fields that do not have the Archimedean Property?

What are examples of ordered fields that do not have the Archimedean Property?
Are the computable numbers one example?