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**Subsets of the set of primes -- uncountable or countable??**

Cantor proved that the sub-sets of the natural numbers are uncountable.

assuming that the the set of primes can be put in a 1-to-1 matching with the natural numbers (which I believe they can...) then it would follow that the sub sets of the set of primes is uncountable.

However, each sub set of the set of primes can be shown to correspond to a unique natural number -- the product of the subsets elements. For, each natural number has a unique prime factorization.

If the sub-sets of the set of primes can be put in a 1-to-1 matching with a a set of numbers that are all natural, clearly this set of numbers that are natural can be put in a 1-to-1 matching with the set of natural numbers, indicating that the subsets of the set of primes are countable

So are the subsets of the set of primes countable or not?

Thanks for reading.