Given a set of countable sets , such that is totally ordered by inclusion, videlicet for every either or . Intuitively, for at every step in this chain one element at least must be added, one expects the set to be countable as well.
Suppose is countable. Then the union, is a countable union of countable sets, hence countable. (Suppose is an enumeration of and enumerations of the elements of the chain. Then enumerates .)
Thus is an upper bound of . In the poset of countable subsets of some set , of which is a subset, every non-empty chain has an upper bound. Hence, using Zorn’s lemma there is a maximal element, say .
Suppose is uncountable, then there exists a . is most definitely also countable and which contradicts ’s maximality. We are forced to conclude that there exists an uncountable chain of countable sets.
Cantor’s set theory keeps surprising.
: an example of such a chain is the set of the countable ordinals.
: a “more concrete” example are the downsets in without the empty set and itself. These downsets correspond to real numbers, see Dedekind Cuts.