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3
votes
1answer
91 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 ...
4
votes
1answer
93 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 ...
3
votes
1answer
67 views

Compatible total orderings of the group $\mathbb{Z}^\mathbb{N}$

Given the additive group of the module $\mathbb{Z}^\mathbb{N}$ and a total ordering of the group that is compatible with addition and where $\chi_{\{n\}} > 0$ for all $n \in \mathbb{N}$, can we say ...
2
votes
1answer
141 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 ...
17
votes
1answer
528 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 ...
22
votes
1answer
422 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
2answers
387 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 ...
30
votes
1answer
2k 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 ...
7
votes
2answers
664 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
1answer
87 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 $(k,P)$ ...
34
votes
6answers
1k 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 ...
5
votes
0answers
530 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
2answers
725 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 ...
9
votes
0answers
239 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 ...
6
votes
4answers
1k 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 ...