Editing Cardinal number (section)

=== Successor cardinal ===
{{Further|Successor cardinal}}

If the axiom of choice holds, then every cardinal κ has a successor, denoted κ<sup>+</sup>, where κ<sup>+</sup> > κ and there are no cardinals between κ and its successor.  (Without the axiom of choice, using [[Hartogs number|Hartogs' theorem]], it can be shown that for any cardinal number κ, there is a minimal cardinal κ<sup>+</sup> such that <!-- κ<sup>+</sup> ࣞ κ.<ref group=notes>The symbol is the [[Unicode]] symbol for not less than or equal to.</ref>--><math>\kappa^+\nleq\kappa. </math>)  For finite cardinals, the successor is simply κ + 1.  For infinite cardinals, the successor cardinal differs from the [[successor ordinal]].