# Set of countable ordinals

1. Mar 12, 2005

### zefram_c

I can't get a good intuitive grasp on this set. Folland defines it as follows:

My questions / problems with it are somewhat as follows:

1) I don't see how the set can be uncountable if any initial segment is countable.

2) How do we know that the element x0 used in the proof exists?

2. Mar 12, 2005

### Hurkyl

Staff Emeritus
If you want an analogy, consider the first infinite ordinal, $\omega$. While it is an infinite ordinal, but every initial segment is finite... if you can understand this, you should be able to understand the first uncountable ordinal.

x0 exists because ordinals are well ordered. Any collection of ordinals has a smallest element.

3. Mar 13, 2005

### mathwonk

Did you understand Hurkyl's response to your question?

I.e. either you meant to ask: why does it follow logically that the statements given are true? (which is an immediate consequence of their definitions and of logic)

or you meant to ask: "I understand the proof, but how can this be possible?"