Alfred Tarski, in his paper "Ueber unerreichbare Kardinalzahlen" Fund. Math. vol 30 (1938) pp 68-89 proves the followig theorem of ZFC "If the cardinal of the set Y is equal to the cardinal of the set of the subsets of Y that are not equipotent with Y, then the cardinal of Y is (strongly) inaccessible". The proof of the paper is rather long and involved. Question: Is there another known simpler proof of this theorem ? Gérard Lang
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In modern notation, it says, "if $\kappa$ is a cardinal and $\kappa ^{< \kappa} = \kappa$, then $\kappa$ is strongly inaccessible." This isn't entirely true since the antecedent holds for $\kappa = \omega$ but $\omega$ isn't considered strongly inaccessible, but that's not a big deal. More importantly, under CH the antecedent will hold of $\aleph _1$ but $\aleph _1$ isn't a limit cardinal. So we need to add the assumptions that $\kappa$ is an uncountable limit cardinal. Given that, we can proceed:
So let's assume $\kappa ^{< \kappa} = \kappa$. First we show $\kappa$ is strong limit: $\kappa \leq 2^{< \kappa} \leq \kappa ^{< \kappa} = \kappa$. Next we show $\kappa$ is regular: Suppose not, then $\kappa ^{< \kappa} = \kappa < \kappa ^{ \mathrm{cf} ( \kappa) } \leq \kappa ^{< \kappa}$, contradiction.
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$\begingroup$ Thank you very much. This is the right answer for my very question ! Gérard Lang $\endgroup$Gérard Lang– Gérard Lang2011-01-19 21:48:01 +00:00Commented Jan 19, 2011 at 21:48
$\aleph_1$has only$\aleph_1$countable subsets. $\endgroup$