TITLE: A question about "small" uncountable cardinal numbers
QUESTION [5 upvotes]: If $X$ denotes a set, let $C(X)$ denote its cardinal number and let $P(X)$ denote its power set. There is a school of thought which considers any set having the cardinal number $C(P(\mathbb{R}))$—where $\mathbb{R}$ denotes the set of real numbers—to be too large for the intuition to grasp. This school mantains that in almost all branches of mathematics except set theory itself, there is no need to require the existence of any set whose cardinal number is greater than $C(\mathbb{R})$.
Although this viewpoint sounds plausible, I wonder whether it might lead to problems in Measure Theory. Suppose one wants to prove in $\mathsf{ZFC}$ that there  exist sets of real numbers which are not Lebesgue measurable. Does there exist any set $M$, definable in $\mathsf{ZFC}$, which satisfies the following conditions:

$M$ is a subset of $P(\mathbb{R})$,
It is provable in $\mathsf{ZFC}$ that at least one element of $M$ is not Lebesgue measurable, and
The cardinal number of $M$ is $C(\mathbb{R})$?

I know a number of examples of sets—definable in ZFC—that satisfy (1) and (2), but none of them also satisfies (3).

REPLY [10 votes]: This is independent. See: Harvey Friedman, On definability of nonmeasurable sets, Canad. J. Math. 32 (1980), no. 3, 653--656.