My main question is: Do the constructable sets of numbers described below have the same density and distribution as $\mathbb P$? Secondarily: Can this be proved, and if so, what (if anything) does the existence of a large number of distinct such sets say about the distribution of numbers in any one such set (such as $\mathbb P$ itself)?
The Sieve of Eratosthenes might be said to operate as follows: $2$ is the smallest prime number, so remove $1$ from consideration. Discard from $\mathbb N$ all numbers that are $>2\ \land\ \equiv 0 \bmod 2$. Move to the next larger prime number and repeat the process. In general, for the $n$th iteration, discard from $\mathbb N$ all numbers that are $>p_n\ \land\ \equiv 0 \bmod p_n$. The choice of the residue class $0$ to be universally excluded is acceptable within the rule (aside from the fact that it yields an especially desired set) because every prime number has a residue class $0$.
But no residue class is privileged; we can make a similar set using the residue class $1$ with respect to appropriate primes. This time, remove $1,2$ from consideration. For the $n$th iteration, discard from $\mathbb N$ all numbers that are $>p_n+1\ \land\ \equiv 1 \bmod p_n$. A little reflection shows that this sieve would yield the set $\{(p_i+1)\}$, which plainly has density and distribution properties identical to $\mathbb P$. Here also, the residue class $1$ to be universally excluded is apt because every prime number has a residue class $1$.
This method of sieving can be expanded to include rules for discarding numbers where the residue class to be excluded is not constant at each iteration, but varies with the prime. For example, for each prime $p_i$, choose $0\le r_i \le p_i-1$. Then the rule for sieving would be: For the $n$th iteration, discard from $\mathbb N$ all numbers that are $>p_n+r_n\ \land\ \equiv r_n \bmod p_n$.
The number of distinct ways of executing such a sieve increases as $p_n\#$, since there are $p_i$ ways to choose $r_i$ for each prime. This fact is sufficiently daunting that I have not tried to look at empirical examples. The results obtained in any one (or few) cases might or might not be reflective of what happens in general. In every case, by the nature of the sieve, the fraction of numbers remaining (i.e. undiscarded) is expected to be $\prod_i (1-\frac{1}{p_i})$. Do sets generated in this manner have the density and distribution properties of $\mathbb P$? Is there a theoretical framework within which this sort of question can be approached?