# Questions tagged [ordered-fields]

The tag has no usage guidance.

44 questions
Filter by
Sorted by
Tagged with
104 views

• 2,484
323 views

• 586
1 vote
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 $x' \in [x-1;x+1]_L$ ...
• 2,417
178 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
273 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 ...
• 2,417
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 $F$?
• 449
176 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 ...
• 2,417
193 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 ...
• 2,417
272 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 ...
• 2,417
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 ...
• 6,621
701 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 ...
• 6,621
574 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 ...
• 41k
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 ...
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 ...
108 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 $(k,P)$ ...
376 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 ...
• 1,406
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 ...
• 64.9k
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
746 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 (...
• 19.5k
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 ...
• 19.5k
314 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 ...
• 19.5k
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 ...
• 19.5k