Questions tagged [ordered-fields]

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59 votes
8 answers
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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 ...
Kevin Buzzard's user avatar
35 votes
6 answers
3k 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 ...
Pete L. Clark's user avatar
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 ...
user avatar
27 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 ...
user107952's user avatar
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23 votes
1 answer
696 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 ...
Asaf Shachar's user avatar
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21 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 ...
Asaf Shachar's user avatar
  • 6,611
18 votes
3 answers
636 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 ...
Uriya First's user avatar
  • 2,846
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 ...
James Propp's user avatar
  • 19.4k
12 votes
2 answers
537 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 ...
Christopher King's user avatar
11 votes
3 answers
761 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 $F$-...
Denis Nardin's user avatar
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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 ...
Zbigniew Fiedorowicz's user avatar
10 votes
2 answers
1k 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 ...
James Propp's user avatar
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10 votes
2 answers
2k views

What fields can be used for an inner product space?

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 appeared here, ...
Stefan's user avatar
  • 523
10 votes
0 answers
313 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 ...
James Propp's user avatar
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9 votes
1 answer
449 views

Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?

We work in ZFC. Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$. A field $E$ is ...
Jakobian's user avatar
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8 votes
2 answers
939 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 ...
tparker's user avatar
  • 1,243
8 votes
2 answers
573 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 ...
Taras Banakh's user avatar
  • 40.8k
7 votes
2 answers
348 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 ...
JOSE MANUEL GAMBOA MUTUBERRIA's user avatar
7 votes
0 answers
98 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 ...
Monroe Eskew's user avatar
  • 18.1k
6 votes
1 answer
243 views

Archimedean ordered fields without maxima and minima in constructive mathematics

In constructive mathematics, let us define an ordered (Heyting) field $F$ to be a commutative ring with a binary relation $<$ which is irreflexive, where for all $x$, $\neg (x < x)$ asymmetric, ...
Madeleine Birchfield's user avatar
6 votes
1 answer
316 views

Proper definition of ordered field in constructive mathematics

The nLab article on ordered fields defines ordered fields to be a field $K$ with a strict linear order $<$ such that $0 < 1$ and for all elements $a \in K$ and $b \in K$, if $a > 0$ and $b &...
Madeleine Birchfield's user avatar
6 votes
0 answers
96 views

Archimedean ordered field in which every function is smooth

In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \...
Madeleine Birchfield's user avatar
5 votes
5 answers
4k 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?
Matthew 's user avatar
5 votes
1 answer
399 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 ...
Keshav Srinivasan's user avatar
5 votes
1 answer
375 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 ...
user38200's user avatar
  • 1,446
5 votes
1 answer
177 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 ...
Twiffy's user avatar
  • 155
5 votes
1 answer
191 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 ...
nombre's user avatar
  • 2,307
5 votes
0 answers
740 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 (...
James Propp's user avatar
  • 19.4k
4 votes
2 answers
357 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 $F$?
Haidar's user avatar
  • 449
4 votes
1 answer
318 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,\...
ar.grig's user avatar
  • 1,133
4 votes
1 answer
272 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 ...
nombre's user avatar
  • 2,307
4 votes
1 answer
137 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 ...
Captain Lama's user avatar
4 votes
0 answers
251 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 ...
user avatar
4 votes
1 answer
249 views

Constructing ordered fields with lattice structure from ordered fields without lattice structure, and vice versa, in constructive mathematics

This post originated from my reference request for the definition of an ordered field in constructive mathematics: Proper definition of ordered field in constructive mathematics We are working in ...
Madeleine Birchfield's user avatar
3 votes
1 answer
170 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 ...
Alec Rhea's user avatar
  • 8,977
3 votes
1 answer
175 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 ...
nombre's user avatar
  • 2,307
2 votes
1 answer
268 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 ...
nombre's user avatar
  • 2,307
2 votes
0 answers
113 views

Can global fields be defined as certain topological fields like local fields?

It's known that local fields can be defined as a non-discrete, Hausdorff (equivalently non-indiscrete), locally compact, topological field, which is the same as non-trivial (i.e. neither discrete nor ...
Z Wu's user avatar
  • 340
2 votes
0 answers
106 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: ...
ΑΘΩ's user avatar
  • 121
2 votes
0 answers
100 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 ...
nombre's user avatar
  • 2,307
1 vote
1 answer
192 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. $\...
Chilote's user avatar
  • 586
1 vote
1 answer
77 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 $x' \in [x-1;x+1]_L$ ...
nombre's user avatar
  • 2,307
0 votes
1 answer
106 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 $(k,P)$ ...
mathcounterexamples.net's user avatar
0 votes
0 answers
86 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 ...
Bib-lost's user avatar
  • 267