Let's say I have an infinite set $\mathcal{P}$ of primes, and I define the set $$\mathcal{S} = \{n \in \mathbb{N}: \; \mathrm{all} \; \mathrm{prime} \; \mathrm{factors} \; \mathrm{of} \; n \; \mathrm{come} \; \mathrm{from} \; \mathcal{P}\}.$$
I would like to relate the asymptotic densities of $\mathcal{P}$ (among primes) and $\mathcal{S}$ (among natural numbers); recall that these are $$\delta(\mathcal{P}) = \lim_{x \rightarrow \infty} \frac{\#\{p \in \mathcal{P}, \; p \le x\}}{\#\{p \, \mathrm{prime}, \; p \le x\}}$$ and $$\delta(S) = \lim_{x \rightarrow \infty} \frac{\#\{n \in S, \; n \le x\}}{\#\{n \in \mathbb{N}, \; n \le x\}}.$$
The specific questions: if $\mathcal{P}$ has zero density, does $\mathcal{S}$ necessarily have zero density? Or is it possible that the extreme opposite is true: does $\mathcal{S}$ have to have nonzero density just by virtue of $\mathcal{P}$ being infinite?