Questions tagged [ordered-fields]
The ordered-fields tag has no summary.
49 questions
7 votes
1 answer
324 views
Example of an $\alpha_2$-$\eta_1$-set with no $(\aleph_1,\aleph_1)$-gaps
My education is mostly in general topology, so forgive me if this is obvious for set theorists. I'm starting to learn more about hyper-real fields (I'm reading Super-real fields by Dales and Woodin), ...
- 2,267
6 votes
1 answer
331 views
What makes the surreals special among other surreal-like fields?
Pre-setup: Let $\kappa = \aleph_\xi$ be an uncountable regular cardinal. Its role in this question is merely to sidestep the technical difficulties surrounding the (imho quite uninteresting) notion ...
- 38k
3 votes
0 answers
67 views
Computing the truncations (“ancestors”) of a surreal number from its Hahn series representation (“normal form”)
If $x$ is a surreal number and $\alpha$ an ordinal, let us denote $T_\alpha(x)$ and call $\alpha$-truncation of $x$ the surreal number whose sign sequence is obtained by truncating the sign sequence ...
- 38k
4 votes
1 answer
234 views
Automorphism on the hyperreals
$\DeclareMathOperator\hal{hal}$A field isomorphism $\phi:F\rightarrow G$ is a bijection such that (i) $\phi(x+y)=\phi(x)+\phi(y)$ and (ii) $\phi(xy)=\phi(x)\phi(y)$, where $F$ and $G$ are ordered ...
- 151
0 votes
0 answers
124 views
Functions representing all strings somewhere
Do there exist "nice" (maybe analytic?) functions $f_0,f_1:\mathbb R \to \mathbb R$ such that $\forall n\in\mathbb N,\forall \sigma\in\{0,1\}^n,\exists x\in\mathbb R, \forall \tau\in\{0,1\}^...
- 25.2k
6 votes
0 answers
134 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} \...
user483446
6 votes
1 answer
349 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, ...
user483446
13 votes
1 answer
822 views
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC?
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 field-isomorphic to $\mathbb{R}$. For example,...
- 2,267
3 votes
0 answers
175 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 ...
- 632
4 votes
1 answer
324 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 ...
user483446
6 votes
1 answer
410 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 &...
user483446
4 votes
1 answer
384 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,\...
- 1,153
4 votes
1 answer
190 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 ...
- 284
7 votes
0 answers
135 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 ...
- 20.7k
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 ...
- 2,183
5 votes
1 answer
540 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 ...
- 4,573
2 votes
0 answers
133 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: ...
- 121
12 votes
2 answers
672 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 ...
- 7,130
0 votes
0 answers
119 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 ...
- 277
18 votes
3 answers
742 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 ...
- 3,198
60 votes
8 answers
9k 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 ...
- 42k
11 votes
3 answers
863 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 $F$-...
- 16.8k
9 votes
2 answers
1k 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 ...
- 1,321
3 votes
1 answer
228 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 ...
- 10.4k
7 votes
2 answers
417 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
133 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 ...
- 2,755
5 votes
0 answers
286 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 ...
user5810
1 vote
1 answer
255 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. $\mathbb{...
- 636
1 vote
1 answer
114 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 $x' \in [x-1;x+1]_L$ ...
- 2,755
5 votes
1 answer
219 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 ...
- 155
5 votes
1 answer
310 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 ...
- 2,755
4 votes
2 answers
417 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 $F$?
- 449
3 votes
1 answer
223 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 ...
- 2,755
5 votes
1 answer
246 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 ...
- 2,755
2 votes
1 answer
333 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 ...
- 2,755
21 votes
1 answer
2k 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 ...
- 6,841
23 votes
1 answer
781 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 ...
- 6,841
8 votes
2 answers
665 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 ...
- 44k
41 votes
2 answers
4k 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 ...
user21349
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
163 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 $(k,P)$ ...
- 1,375
5 votes
1 answer
428 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 ...
- 1,416
35 votes
6 answers
4k 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 ...
- 66.4k
11 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, ...
- 533
5 votes
0 answers
845 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 (...
- 20.1k
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 ...
- 20.1k
11 votes
0 answers
343 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 ...
- 20.1k
12 votes
5 answers
3k 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 ...
- 20.1k
6 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?
- 71