# Is it useful to think of the natural numbers as a powerset of the primes?

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. –  mjqxxxx Jan 20 '12 at 16:29