<p>This paper has two goals. First, we reconstruct Wittgenstein’s views on what counts as a legitimate irrational – since, as he repeatedly points out, and in agreement with mathematicians such as Emile Borel, not just every infinite string of digits qualifies as one. Once his conception (‘full-blooded intensionalism’) is sketched out, and its specificity is highlighted by comparing it with two other cognate views (‘extensionalism’ and ‘quasi-intensionalism’), our second objective is to examine how his type of intensionalism impacts his attitude towards Cantor’s theorem. In this regard, the more general claim we argue for is that, despite appearances to the contrary, Wittgenstein was not a revisionist about set-theoretical practice.</p>