Talk:Successor cardinal

Successor cardinal and Hartogs number
Isn't the definition provided at the end of current revision (as a definition without AC)
 * $$\kappa^+ = |\inf \{ \lambda \in ON \ |\ |\lambda| \nleq \kappa \}|$$.

the same thing as the Hartogs number?


 * Kompik: Yes, it is. Please sign your contributions to talk with four tildas, e.g. ~ . JRSpriggs 03:51, 30 May 2006 (UTC)

Without choice?
Do we want to include the following results:

If $$\kappa$$ is any cardinal, then $$\kappa^+$$ (Hartogs number) is a smallest cardinal m such that $$\kappa\nleq m. $$ (Already included, in part.)

If $$\kappa $$ Dedekind finite, then $$ \kappa + 1 $$ is the successor cardinal of $$\kappa $$.

I believe I can show that if $$\kappa $$ is Dedekind infinite, then $$\kappa + \kappa^+$$ is a successor cardinal of $$\kappa $$, but I don't have a reference.

Hence, any cardinal has a successor cardinal. — Arthur Rubin (talk) 18:32, 22 June 2015 (UTC)