The other answers are excellent. But since you have adopted such a strong notion of definability, let me augment them with a positive observation.

The fact is that *any* particular set can be made definable without parameters in a forcing extension of the universe $V$. Indeed, there is a single definition $\varphi(x)$, such that for any set $A$ at all, there is a forcing extension $V[G]$ in which $A$ is the unique set such that $\varphi(A)$. Furthermore, one can arrange that the forcing extension $V[G]$ agrees with $V$ far beyond the reals, so that it has the same reals, the same sets of reals, the same measurable sets and so on for quite a long way.

In particular, there is a single definition such that for any non-Lebesgue measurable set $A$ that you favor, there is a forcing extension $V[G]$, an alternative set-theoretic universe, in which $A$ is defined by $\varphi$ and still non-measurable there.

Let me explain the proof. Fix any set $A$. Let $\kappa$ be the cardinality of the transitive closure $\text{TC}(\{A\})$. Thus, there is binary relation $E$ on $\kappa$ for which $\langle\kappa,E\rangle\cong\langle\text{TC}(\{A\}),{\in}\rangle$. This isomorphism is unique, since it is precisely the Mostowski collapse. Let $E_0\subset\kappa$ be the set of ordinals coding pairs in $E$. In the style of Easton's theorem, let $\mathbb{P}$ be the forcing notion coding the GCH pattern on the regular cardinals above $2^{\aleph_0}$ to first have a block of length exactly $\kappa$ on which the GCH holds, and then an violation of GCH and then a sequence of length $\kappa$ on which the GCH pattern on the regular cardinals matches the elements of $E_0$. In the resulting forcing extension $V[G]$, the cardinal $\kappa$ and the set $E_0$ and hence $E$ and hence $A$ are definable without parameters. Because the forcing is sufficiently closed, it does not adds new reals or sets of reals and it does not affect measurability. So in the extension, the set $A$ is definable by the formula $\varphi$ that expresses the decoding of the GCH pattern to $E_0$ and hence $E$ and hence $A$. This coding idea is due originally to K. McAloon.

The conclusion is that there is a kind of universal definition $\varphi$, which can serve to define any object at all, if only you apply the definition in the correct set-theoretic universe.

10, 1927, p. 77), see also planetmath.org/encyclopedia/… $\endgroup$ – Theo Buehler Jul 3 '11 at 20:39