I am confused about the concept of the “first uncountable ordinal.” Here are 3 statements that are apparently equivalent:
(1) There is an uncountable ordinal that is “first” in the sense that it is the smallest ordinal that, when considered as a set, is uncountable.
(2) This ordinal is the supremum of all countable ordinals.
(3) The elements of the set are the countable ordinals, of which there are uncountable many.
First, which statement is generally given as the definition of “first uncountable ordinal”?
Secondly, how do we know the supremum in statement (2) exists?
Third, why does statement (3) follow from the other statements? And vice versa?