set of countable ordinals  by zefram_c Tags: countable, ordinals
zefram_c is offline Mar12-05, 03:06 PM P: 260 I can't get a good
intuitive grasp on this set. Folland defines it as follows:
Quote Quote by Folland, Real Analysis There is an uncountable
well ordered set S such that Ix={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. Ix0 is
uncountable, in which case let S be Ix0 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? Phys.Org
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