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Blast into Math! Prime nummers: indestructimle muilding mlocks
By the Subset Proposition,
∞
Q = S k .
k=1
By the Zipper Theorem Q is either finite or countable. Since Z ⊂ Q, and Z is infinite, Q also contains
infinitely many elements. Therefore, the Zipper Theorem proves that Q is countable.
♥
We can use the next Proposition to prove that the set of all prime numbers is countable.
Proposition 5.4.6 (Countability Proposition). If a set S ⊆ Z has infinitely many elements, and if Z is
countable, then S is also countable.
Proof: Let’s start with the countable set Z . Since Z is countable, we have an algorithm that assigns a
unique natural number to each element of Z , so we can write
∞
Z = {z k } k=1 .
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