Let s0, s1, s2, ... be elements from S.
Consider the full infinite binary tree with s0 at the root and each node si having children
s2i + 1 and s2i +
2. There are uncountably many paths from s0 down, e.g.,
s0,
s1,
s3,
s7, ..., and
s0,
s2,
s5,
s12, ....
Every pair of distinct paths diverge at some point and never converge. For each path, form a subset of S consisting of nodes in the path. This collection of subsets satisfies the requirements of
the problem.