Set of Sets can be Defined as Family
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Theorem
Let $\Bbb S$ be a set of sets.
Then $\Bbb S$ can be defined as an indexed family of sets.
Proof
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
and:
- the set indexed by itself.
It follows that each of the sets $i \in \Bbb S$ is both:
- an index
and:
- a term $S_i$ of the family of elements of $\Bbb S$ indexed by $\Bbb S$.
Thus we would write $\Bbb S$ as:
- $\family {S_i}_{i \mathop \in \Bbb S}$
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 9$: Families