Understanding Uncountable Well-Ordered Sets: An Intuitive Explanation

[zefram_c]

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

There is an uncountable well ordered set S such that I_x={y in S: y<x} is countable for each x in S. If S' is another set with the same properties, then S and S' are order isomorphic.

Proof: Uncountable well-ordered sets exist by the WEP, let X be one. Either X has the desired property or there is a minimal element x₀ s.t. I_x0 is uncountable, in which case let S be I_x0

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?

 

[Hurkyl]

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.

 

[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?"

Hurkyl answered the second version of your question.

re-reading, it seems your question 1) was the psychological one,
and your 2) was a (tauto)logical one.

ah yes, you said you wanted an intuitive grasp of the set, so Hurkyl understood you correctly.

 

[zefram_c]

Thanks Hurkyl - after reading your response, I went back and checked. Turns out my understanding of well-ordering was incorrect; I now get the logic. The analogy gave me food for thought, so I will think on it for now and see if I can persuade myself that it works.