# If $\kappa$ is infinite and $\kappa$ is not a sum of $<\kappa$ cardinals each less than $\kappa$, then $\kappa$ is regular.

An infinite cardinal $$\kappa$$ is regular if $$\mathrm{cf}(\kappa) = \kappa$$.

It is known that if $$\kappa$$ is regular, then for any family $$(\kappa_i)_{i \in I}$$ of cardinals $$\kappa_i < \kappa$$ with $$|I| < \kappa$$ $$\kappa \neq \sum_{i \in I} \kappa_i$$.

I want to prove the converse: let $$\kappa$$ be an infinite cardinal; if for any family $$(\kappa_i)_{i\in I}$$ of cardinals $$\kappa_i < \kappa$$ with $$|I| < \kappa$$ we have $$\kappa \neq \sum_{i \in I} \kappa_i$$, then $$\kappa$$ is regular.

One should possbily proceed by contradiction. Assume that $$\mathrm{cf}(\kappa) < \kappa$$. Then there is an ordinal $$\alpha < \kappa$$ for which there is a cofinal map $$f\colon\alpha\to\kappa$$, that is, a map $$f\colon \alpha\to\kappa$$ so that $$(\forall \beta < \kappa)(\exists \gamma < \alpha)(\beta \leq f(\gamma))$$. From this we should somehow derive a contradicting by showing that $$\kappa$$ is equal to $$\sum_{i \in I} \kappa_i$$ for some family $$(\kappa_i)_{i \in I}$$ of cardinals.

There really is only one option that sticks out: $$\sum_{\gamma\in\alpha}f(\gamma)$$
• Can you elaborate on why $\bigcup_{\gamma < \alpha} f(\gamma) = \kappa$? Indeed, $(\forall \beta < \kappa)(\exists \gamma < \alpha)(\beta \leq f(\gamma))$. Note that inequality $\leq$ is not strict (in was in my original question, an unfortunate typo) since it is what the definition of the a cofinal map requires. Sep 24, 2018 at 16:44
• @Jxt921 The strict inequality is a slight inconvenience but doesn't change anything really. If $\beta<\kappa$ then $\beta+1$ is still smaller than $\kappa$, and you can find a $\gamma<\alpha$ such that $\beta+1\leq f(\gamma)$ and therefore $\beta<f(\gamma)$. Now note that $\sum f(\gamma)$ cannot be smaller than $\kappa$ because any smaller ordinal is surpassed by some term in the sum. And it cannot be larger because $\sum f(\gamma)\leq\sum \kappa=\kappa$. Sep 24, 2018 at 19:38