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Dec 30, 2014 at 15:56 comment added Thomas Benjamin Actually, I should really say that evolving mathematical practice might force one to construct a new Vitali set not in $L$--sorry. As for the 'evolving continuum' one could always form new Vitali sets by adding reals so as not to violate CH, right?
Dec 30, 2014 at 15:34 comment added Thomas Benjamin @Prof. Hamkins: Thank you for that very nice answer--it is very helpful. The case that interests me most is the last one, that is, the case where one forces to add a new Vitali set without adding reals so that one can have $ZFC+GCH$+$V$$\neq$$L$ with Vitali sets in both $L$ and $V-L$. In this case, is there a way to distinguish the Vitali sets in $L$ from those in $V-L$? I ask this question to see if it makes sense to speak of an 'evolving continuum'--that is, a continuum that grows larger to meet the needs of evolving mathematical practice.
Dec 30, 2014 at 15:18 vote accept Thomas Benjamin
Dec 30, 2014 at 14:24 history answered Joel David Hamkins CC BY-SA 3.0