The Math-tea Argument (i.e. the argument that, for example, there must be real numbers that we cannot describe or define, because there are only countably many definitions, but uncountably many reals) pops up in some very interesting places. Consider the following statement from Prof. Alexandre Miquel of the Universidad De La Republica, Uruguay, found in his course notes titled "An axiomatic presentation of the method of forcing" under the subheading "The fundamental under-specification of the powerset" (the reader can find these course notes under title on the Web):
"The reader might wonder how it is possible to add new subsets to already existing infinite sets, and in particular to the set $\omega$ of natural numbers. Mathematicians are so used to manipulate subsets of $\omega$ (or real numbers) that they often develop the (wrong) intuition that all these subsets are given a priori.
But one should remember that the language we use to define subsets of $\omega$ is (intuitively) denumerable, so that the (intuitive) collection formed by all the subsets of $\omega$ that we can individually define is denumerable. The vast majority of the subsets of $\omega$, that only exist via Cantor's diagonal argument, are definitely out of reach by linguistic means--at least individually--and their existence is, in some sense, purely formal. The method of forcing precisely exploits this fundamental under-specification of the powerset (of infinite sets) by specifying extra subsets by extra linguistic means...."
Is this statement by Prof. Miquel correct, and if not, why not, given the results of Hamkins, Linetsky, and Reitz?