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.

We can recognize the behaviour of minima and maxima in a lot of other things. Take for instance set inclusion as order with intersection as minimum and union as maximum. Actually, the link between general order and set inclusion is frequently made to then propose that “intersection of two set of cases” and “logical and” do look a lot alike.

This is just the tip of the huge iceberg. Order appears everywhere! Everywhere in Math. Everywhere in CS. And its just recognizing simple order I’ve demonstrated. Other useful concepts in order theory that I didn’t even touch are Galois Connections and formal concept analysis.

Oh, another example of an order are integers with bitwise or and bitwise and. It is left as an exercise to the reader when one integer is greater than another.

Interested? Buy an Introduction to Lattices and Order.