 Cantor’s response to both of these paradoxes was telling (and can be formalized in ZFC or in the related systems which admit proper classes, as we will see below). He essentially reinvoked the classical objections to infinite sets on a higher level. Both the largest cardinal and the largest ordinal arise from considering the very largest collections (such as the universe \(V)\). Cantor drew a distinction between legitimate mathematical infinities such as the countable infinity of the natural numbers (with its associated cardinal number \(\aleph_0\) and many ordinal numbers \(\omega , \omega + 1, \ldots ,\omega + \omega ,\ldots)\), the larger infinity of the continuum, and further infinities derived from these, which he called transfinite, and what he called the Absolute Infinite, the infinity of the collection containing everything and of such related notions as the largest cardinal and the largest ordinal. In this he followed St. Augustine (De Civitate Dei) who argued in late classical times that the infinite collection of natural numbers certainly existed as an actual infinity because God was aware of each and every natural number, but because God’s knowledge encompassed all the natural numbers their totality was somehow finite in His sight. The fact that his defense of set theory against the Burali-Forti and Cantor paradoxes was subsequently successfully formalized in ZFC and the related class systems leads some to believe that Cantor’s own set theory was not implicated in the paradoxes.