Adam of Balsham (Parvipontanus) writing his Ars Disserendi in 1132 A.D. reaches the notion of a set A equinumerous with a set B containing A as a proper sub-set. References by page and line are to [Minio-Paluello].

It can be asked, says Adam (92,13) whether this one is something, and whether that one is something, and these are two askables (interrogabilia), and again whether a third is something, and similarly of anything whatever. He thinks, then, of the totality of ‘objects’ (92,12) as numbered. But this question can be put not only about those which are not askables; of any askable too it can be asked whether it is something. Whether the questions be actually put or not, makes no difference; even when they are not put, the points remain askable. Thus the totality of askables is seen to be equinumerous with the totality of objects, which last comprises both the objects which are not askables and those which are. And so the same set is more numerous than itself.

Conscious that this paradoxical situation may be ascribed to his novel terminology and methods, Adam points out that a similar puzzle arises over things enunciated. Even if you do not want to term them ‘enunciables,’ still they are generally said to be true or false. Take the true ones, and compare them as to number with the totality of objects of which enunciations can be made. There he leaves this second version, but it is easy to see how the same result as before can be reached.

The argument lightly paraphrased in the last paragraph but one concludes with the words ‘which is impossible’ (92,21). Instead, however, of rejecting some of his premisses, Adam lets everything stand, and breaks off the discussion on the ground that his purpose is to expound a finite process of achievement (artis viam finitam) rather than what infinitely exceeds our capability to construct − or perhaps, the infinitely extending realm of possibility (possibilitatis excessum infinitum, 93,3–5).