# Proving countable ordinal embeds in R

1. Feb 2, 2016

### cragar

1. The problem statement, all variables and given/known data
show that if q is any countable ordinal, then there is a countable set A ⊆ R (in fact we can require A ⊆ Q), so that (A, <) ∼= (q, ∈).
3. The attempt at a solution
since q is a countable ordinal this implies that it has a mapping to the naturals.
to me this seems strong enough. and its also well ordered.

2. Feb 2, 2016

### andrewkirk

It's hard to tell from the notation you've used, but I suspect the question is asking you to prove there is an order-preserving mapping.
From countability of q you can get a bijection to the naturals, but in most cases it won't be order preserving.

Last edited: Feb 4, 2016
3. Feb 3, 2016

### cragar

in order to preserve order, can we use a permutation group to create a linear order

4. Feb 3, 2016

### andrewkirk

The permutation groups I am familiar with are on finite sets. Unless q is finite - in which case the entire problem becomes trivial - I can't see how a permutation on a finite number of elements would help. I think you will need to use the structure of countable ordinals to prove this.

5. Feb 4, 2016

### micromass

Staff Emeritus
Use transfinite induction and the fact that $\mathbb{R}$ is order isomorphic to any interval $(a,b)$.