# understanding cardinal numbers arithmetic

I have a question about notation in a book I'm reading on set theory and beside of my question I will be glad for a recommendation for a good book that explains well cardinal numbers arithmetic.

If we define that $\kappa$ is a cardinal number if $\kappa$ is an ordinal number such that for every $\alpha <\kappa$ there is no $f:\alpha\to\kappa$ that is surjective, in what sense for two infinite cardinal numbers $\kappa$ and $\gamma$ , we have that $\kappa +\gamma=\kappa\cdot\gamma=\max \{\kappa,\gamma\}$? Is this ordinal arithmetic? or is it arithmetic in the sense of bijections of disjoint unions and products?

One additional comment: $\kappa \cdot \gamma = \max \{ \kappa,\gamma \}$ requires choice to be proven for general infinite cardinals.
• Ian: $\omega <\omega+1$, and yet $\omega+\omega+1>\omega+1$. What you described are the (infinite) indecomposable ordinals, which coincide with the (infinite) ordinal powers of $\omega$: The ordinals of the form $\omega^\alpha$ with $\alpha\ge1$. – Andrés E. Caicedo Nov 26 '14 at 20:20
• For well-ordered infinite cardinals (as here) that $\kappa+\lambda=\max\{\kappa,\lambda\}$ does not use choice. – Andrés E. Caicedo Nov 26 '14 at 20:23