Tarski's caracterisation of REGULAR cardinals

Question

Alfred Tarski, in his paper "Ueber unerreichbare Kardinalzahlen" Fund. Math. 1938 proves the following ZFC theorem "if the cardinal of the set Y is equal to the cardinal of the set of subsets of Y that are not equipotent to Y, then the cardinal of Y is REGULAR." The proof of the paper is rather long and involved. Question: Is there another known simpler proof of this theorem ? Gérard Lang

Answer 1

Let me prove the contraposition: If $\kappa$ is not regular, then it has more than $\kappa$ subsets of size strictly less than $\kappa$.

Suppose $\kappa$ is not regular. Let $\lambda$ be the cofinality of $\kappa$. Note that the number of subsets of $\kappa$ of size $\lambda$ is $\kappa^\lambda$. It is therefore enough to show that $\kappa^\lambda$ is strictly larger than $\kappa$.

Choose a cofinal sequence $\{\alpha_\gamma:\gamma\in\lambda\}$ of ordinals in $\kappa$. Let $f:\kappa\to{}^\lambda\kappa$ be a function. We show that it is not onto. Define $g:\lambda\to\kappa$ be letting $g(\gamma)$ be the least element of $\kappa\setminus\{(f(\beta)(\gamma):\beta\in\alpha_\gamma\}.$ This exists since $\{(f(\beta)(\gamma):\beta\in\alpha_\gamma\}$ is of size striclty less than $\kappa$. Then $g$ does not appear in the range of $f$.

For suppose $g=f(\alpha)$ for some $\alpha\in\kappa$. Choose $\gamma\in\lambda$ such that $\alpha\in\alpha_\gamma$. Then $g(\gamma)\not\in\{f(\beta)(\gamma):\beta\in\alpha_\gamma\}$ and hence in particular, $g(\gamma)\not=f(\alpha)(\gamma)$, a contradiction.

This is, I believe, the usual proof of König's Theorem (not the tree-lemma) that you should find in any standard text book on set theory.