This following must obviously be too simple, but if someone can explain what it misses, I'd be very grateful.
If there are only finitely many pairs (n, n+2), both prime, then from the apparently infinitely many numbers x such that x is
neither mod2 nor mod2, nor mod3 nor mod3, nor mod5 nor mod5.... nor modp nor [p-2](modp for all p < sqrt(x + 1)),
then above a certain size, either the seeming inductive step for filtering all candidate numbers x by the two requirements modulo the next additional prime breaks down, or else some extra condition begins to hold.
Either way, somehow even if infinitely many x remain satisfying the conditions for each specific p_i from then on, none of them after this point need be small enough to ensure any longer that infinitely many x meet these conditions for all values of p < sqrt(x + 1) in general? Is this the main problem with trying to show this?