Please forward this error screen to sharedip-16015392192. Number of alphabet you are the account owner, please submit ticket for further information. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. 2, ω2, ωω and ε0 are among the countably infinite sets.

This ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Using the axiom of choice we can show one of the most useful properties of the set ω1: any countable subset of ω1 has an upper bound in ω1. This follows from the fact that a countable union of countable sets is countable, one of the most common applications of the axiom of choice. 1 is actually a useful concept, if somewhat exotic-sounding. The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers.

For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal.