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)