Set of Sets/Examples/Set of Initial Segments

Example of Set of Sets
Let $\Z$ denote the set of integers.

Let $\map \Z n$ denote the initial segment of $\Z_{> 0}$:
 * $\map \Z n = \set {1, 2, \ldots, n}$

Let $\mathscr S := \set {\map \Z n: n \in \Z_{> 0} }$

That is, $\mathscr S$ is the set of all initial segments of $\Z_{> 0}$.

Then:
 * $\mathscr S := \set {\set 1, \set {1, 2}, \set {1, 2, 3}, \ldots}$

and we have that:
 * $\mathscr S \subsetneq \powerset \Z$

where $\powerset \Z$ denotes the power set of $\Z$.