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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?

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This is cardinal arithmetic, not ordinal arithmetic, so yes, it's all about existence of bijections related to disjoint unions and products.

One additional comment: $\kappa \cdot \gamma = \max \{ \kappa,\gamma \}$ requires choice to be proven for general infinite cardinals.

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  • $\begingroup$ 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 $\alpha\ge1$. $\endgroup$ – Andrés E. Caicedo Nov 26 '14 at 20:20
  • $\begingroup$ For well-ordered infinite cardinals (as here) that $\kappa+\lambda=\max\{\kappa,\lambda\} $ does not use choice. $\endgroup$ – Andrés E. Caicedo Nov 26 '14 at 20:23
  • $\begingroup$ @AndresCaicedo For your first point, that's valid, I edited accordingly. For your second point, I didn't say anything to the contrary. $\endgroup$ – Ian Nov 26 '14 at 20:42

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