Set of Sets can be Defined as Family

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Let $\Bbb S$ be a set of sets.

Then $\Bbb S$ can be defined as an indexed family of sets.


Let $S: \Bbb S \to \Bbb S$ denote the identity mapping on $\Bbb S$:

$\forall i \in \Bbb S: S_i = i$

where we use $S_i$ to mean the image of $i$ under $S$:

$S_i := \map S i$

Then we can consider $S$ as an indexing function from $\Bbb S$ to $\Bbb S$.

Hence in this case $\Bbb S$ is at the same time both:

an indexing set


the set indexed by itself.

It follows that each of the sets $i \in \Bbb S$ is both:

an index


a term $S_i$ of the family of elements of $\Bbb S$ indexed by $\Bbb S$.

Thus we would write $\Bbb S$ as:

$\family {S_i}_{i \mathop \in \Bbb S}$