An Uncountable Chain

To understand this problem, you must be familiar with the notions of countable and uncountable sets. A countable infinite set is one whose members can be placed into one-to-one correspondence with the natural numbers. The members of an uncountable set, however, cannot be placed into one-to-one correspondence with the natural numbers. Cantor proved that the real numbers are uncountable, before going insane.

A collection of sets is known as a "chain" if it has the property that for any two sets in the collection, one of them is a proper subset of the other.

The problem: Is it possible to construct an uncountable chain of subsets of the natural numbers?

In other words, is there a collection of sets with the following properties:
- the elements of each set are natural numbers
- for any two sets in the collection, one is a proper subset of the other
- the collection contains uncountably many sets

Solution