 One lesson to be learned from this, according to Wittgenstein (RFM II, §33), is that “there are diverse systems of irrational points to be found in the number line”, each of which can be given by a recursive rule, but “no system of irrational numbers”, and “also no super-system, no ‘set of irrational numbers’ of higher-order infinity”. Cantor has shown that we can construct “infinitely many” diverse systems of irrational numbers, but we cannot construct an exhaustive system of all the irrational numbers (RFM II, §29). As Wittgenstein says at (MS 121, 71r; Dec. 27, 1938), three pages after the passage used for (RFM II, §57): 