As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers.
- The lowest transfinite ordinal number is ω.
- The first transfinite cardinal number is aleph-null, , the cardinality of the infinite set of the integers. The next higher cardinal number is aleph-one, .
The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers.
In both the cardinal and ordinal number systems, the transfinite numbers can keep on going forever, with progressively more bizarre kinds of numbers.