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.
Update: an example of such a chain is the set of the countable ordinals.
Another update: a “more concrete” example are the downsets in without the empty set and
itself. These downsets correspond to real numbers, see Dedekind Cuts.