The existence of Sierpiński sets is independent of the axioms of ZFC.
Sierpiński (1924) showed that they exist if the continuum hypothesis is true.
On the other hand, they do not exist if Martin's axiom for ℵ1 is true.
By the continuum hypothesis, it is possible to enumerate them as Sα for countable ordinals α.
For this one modifies the construction above by choosing a real number xβ that is not in any of the countable number of sets of the form (Sα + X)/n for α < β, where n is a positive integer and X is an integral linear combination of the numbers xα for α < β.