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 $\Z$.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.8$. Sets of sets: Example $28$