Timeline for isomorphism of Dedekind complete ordered fields

7 events

when toggle format	what		by	license	comment
Jan 2, 2013 at 21:13	history	edited	Clive Newstead	CC BY-SA 3.0	added 220 characters in body
Jan 2, 2013 at 20:52	comment	added	Clive Newstead		Another complete ordered field is $F = \{ (x,2x)\, :\, x \in \mathbb{R} \} \subseteq \mathbb{R}^2$, which is the line through the origin with gradient $2$. The isomorphism $\mathbb{R} \to F$ is given by $x \mapsto (x,2x)$. We also have copies of $\mathbb{N}, \mathbb{Z}, \mathbb{Q}$ inside $F$ given by, for instance, $\mathbb{N}_F = \{ (n, 2n)\, :\, n \in \mathbb{N}_{\mathbb{R}} \}$, where I've used $\mathbb{N}_{\mathbb{R}}$ to denote the copy of $\mathbb{N}$ in $\mathbb{R}, whatever that is.
Jan 2, 2013 at 20:51	comment	added	Clive Newstead		Ok sure. Say you have $\mathbb{R}$. The natural inclusion $i_{\mathbb{N}} : \mathbb{N} \hookrightarrow \mathbb{R}$ is given by $i_{\mathbb{N}}(n) = n$, where the $n$ on the right-hand side is the element of $\mathbb{R}$ which is '$n$ units along the number line'. If you're viewing $\mathbb{R}$ as the set of Dedekind cuts of $\mathbb{Q}$, then you could set $i_{\mathbb{N}}(n) = \{ q \in \mathbb{Q}\, :\, q < n \}$.
Jan 2, 2013 at 20:47	comment	added	Chris		Maybe, inducting using the natural inclusion?
Jan 2, 2013 at 20:41	comment	added	Clive Newstead		An example of what? A complete ordered field?
Jan 2, 2013 at 20:36	comment	added	Chris		Could you give me an example (maybe, one of the cases Spivak wants to prove)?
Jan 2, 2013 at 20:17	history	answered	Clive Newstead	CC BY-SA 3.0