 Consider the reals which are definable in finitely many words. They form a countable sequence: \(E_0, \ldots, E_n,\ldots\). Since the continuum is uncountable, there exist reals not occurring in the given enumeration. Assuming that the continuum is well-ordered, there exists “the least real \(E\) which is not in the sequence \(\{E_n \mid n \in \Omega \}\)”; this real is not in the sequence, but the very expression “the least real \(E\) which is not in the sequence” defines \(E\) with finitely many words; so \(E\) occurs somewhere in the sequence: contradiction!