I find it necessary to be patient with Cantor’s theorem. As elementary as it may be to some (some tell me it is elementary to them and who am I to doubt it), it causes some philosophical intuitions to be changed (even to be reversed, as I like to say) if one gives it the chance by repeated musing. And what else does that? (Has the power to displace invested intuitions, that is?)
I for one am not sure that the theorem is elementary so much as startlingly accessible (indeed inevitable) and inexhaustible. What we should want from a principle.
What is a philosophical interpretation of Cantor’s theorem? There are two cases: “interpretations” that do and do not recognize that the theorem has something to say about interpretation, that is, about the relation of syntax and semantics, including in its own case, as much as that might (should) give us pause.
I am not speaking of silent contemplation, as of a Plotinian mystery, as occupying this pause. Nothing comes from nothing here; one must keep directing philosophical questions to the theorem, but then one must have the patience to let the theorem respond. If one refrains from interrupting it too quickly, i.e. from redirecting the potentiality of the question toward one or another established discourse for possible answers, the theorem regains a measure of sovereignty over the very concepts with which one approaches to question it. It pushes back. One gains in this way – by a sort of question and response (not answer) – access to unexpected depths in the theorem. Of course a theorem does not speak in the manner of an oracle. What one has to learn to look for, in order to let the theorem answer, are the metalogical dimensions of the questions one puts to it. Reversal is enacted by finding that the concept of each question, far from simply framing the theorem, is in part already inscribed in it. Cantor’s theorem (or read: diagonalization) is the form of which the concept posed about it is always-already in question in it. This reversal of the anticipated priority of concepts is the theorem’s answer, its teaching of philosophy, and its retrenchment of itself at the level of a philosophical principle, relative to the concepts one brings to it.
(This note does not escape the emplacement it thematizes, nor would I wish it to. It would be possible, fruitful, to expand on what the form has to say about the somewhat phenomenological themes that have fallen out, partly by chance, in this train of thought – including the relation of instrumental rationality to another thinking, potentiality vs. possibility, and the adumbration of at least two levels of patience, the easier pertaining to the theorem itself which reverses the question, and the more difficult, commanded of thinking by it toward what may withdraw from the question: certain phenomena and events, certain other/s.)