Proving countable ordinal embeds in R

[cragar]

Homework Statement

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, ∈).

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.

 

[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.

 

[cragar]

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

 

[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.

 

[micromass]

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