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As a consequence of the fundamental theorem of arithmetic, it seems that the powerset of the prime numbers uniquely identifies each natural number, $\mathbb{N_1}=\mathcal{P}(\mathbb{P})$ (here I'm assuming that the empty set corresponds to $1$). As someone who is still making the connections in mathematics, it would be of interest to know if this is a useful construction of the natural numbers - and if so, to what purpose?

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    Square-free supernatural numbers are isomorphic to the power set of the primes: the union and intersection operations on the latter correspond precisely to the least-common-multiple and greater-common-divisor operations on the former.2012-01-20

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This is false for two reasons: (a) infinite sets of primes do not determine a natural number; (b) your proposed representation can not account for natural numbers with a repeated prime factor.

Here are two statements which are true:

The set of square-free natural numbers corresponds naturally to the set of finite sets of primes.

The set of all natural numbers corresponds naturally to the set of finite multi-sets of primes.

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    @Henning A [supernatural number.](http://en.wikipedia.org/wiki/Supernatural_numbers)2012-01-20