 The founder of transfinite arithmetic, Georg Cantor (1845–1918), is also a founding father of set theory. He famously proved that the set of real numbers has a larger cardinal number than the set of natural numbers; the set of reals has the same cardinality as the power set (the set of all subsets) of the set of naturals. Cantor further argued that \(\aleph_0\) is the first (and smallest) transfinite cardinal number in an infinite series of increasingly larger transfinite cardinals, \(\aleph_0,\) \(\aleph_1,\) \(\aleph_2\), and so on. But note that the numerical subscripts of these alephs do not refer to cardinal numbers, but rather refer to ordinal (or ordering) numbers, e.g., first, second, third, and so forth. According to Cantor, there is no largest transfinite cardinal number, and although there are infinitely many transfinite cardinal numbers, there is no cardinal number of them. Rather, the totality of transfinite cardinals is absolutely infinite in a sui generis non-arithmetical sense. Similarly, in Zermelo-Fraenkl set theory (ZF), and its extension ZFC (ZF + the axiom of choice), the totality of transfinite cardinal numbers does not qualify as a set having a definite cardinal number of members. Such a set would be too large; were such a set to exist, paradoxical consequences would ensue akin to Russell’s paradox. ZF avoids these paradoxical consequences by characterizing the totality of transfinite cardinals as an infinitely large proper class. Within the forms of axiomatic set theory that have attracted significant support it cannot be proved that one proper class is larger than another.