Set of subsets of naturals with inclusion order [Solution]

The goal was to find an uncountable set $S$ such that each member of $S$ was a subset of the naturals, and given any two elements $A, B \in S$, either $A \subset B$ or $B \subset A$.

Solution

Consider the rationals in $[0, 1]$ which are countable, say $\{q_1, q_2, \dots\}$

For each real $x \in [0, 1]$ let $S_x = \{i : q_i \le x\}$.

$S = \{S_x : x \in [0, 1]\}$ is a set with that property.

Set of subsets of naturals with inclusion total order

Can you find an uncountable set $S$ with the following properties?

- every member of $S$ is a subset of the natural numbers.
- for any $A \in S$, $B \in S$ either $A$ is a subset of $B$, or vice-versa.