Page 109 - 'Blast_Into_Math

Blast into Math! Prime nummers: indestructimle muilding mlocks

Theorem 5.4.4 (Zipper Theorem). If each set S k is finite and not empty, then

∞

S = S k
k=1
is either countable or finite.

Proof: If

∞

S = S k
k=1

contains finitely many elements, then there is no work to be done, because it is finite. If S contains
infinitely many elements, then we need a way to assign precisely one natural number to each element of
S . How can we find the first element of S ? Well, let’s start with S 1 . Since S 1 is finite and is not empty,
there is some s 1 ∈ S 1 . Let’s define

z 1 = s 1 .
Now, either

S 1 \{z 1 } = ∅

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