On Splitting Numbers(Mathematical Logic and Applications'92)
Toshio Suzuki · Institutional Repositories DataBase (IRDB) · 1993
We shall discuss about splitting numbers of uncountable regular cardinals.1. Question about splitting numbers The author introduced a question about splitting numbers of uncount- able regular cardinals at a talk in Aug. 4. 1992 at RIMS, Kyoto.Let $\kappa$ be an infinite cardinal.$S\subseteq[\kappa]^{\kappa}$ $(=\{X\subseteq\kappa : |X|=\kappa\})$ is called a splitting family on $\kappa$ if for all $X\in[\kappa]^{\kappa}$ there exists an $A\in S$ such that $|X\cap A|=|X\backslash A|=\kappa$ .The splitting number of $\kappa$ is the minimum cardinality of a splitting family on $\kappa$ .We denote it by $s(\kappa)$ .In particular, $s(\omega)$ is the original splitting number, which is one of the so-called six cardinals and the following is well-known.FACT.[2](1) $s(\omega)\geq\omega_{1}$ .(2) $Con(ZFC)impli$es $Con(ZFC+s(\omega)\geq\omega_{2})$ .What about uncountable regular cardinals ?In 1991, M. Motoyoshi studied splitting numbers of uncountable regular cardinals under the su- pervision of S. Kamo.FACT.[5] Let $\kappa$ be an uncounta $ble$ regu1ar cardinal.Then rc $is$ $st$ rongly in $accessible\Leftrightarrow s(\kappa)\geq\kappa$ .At that time I pointed out the following fact.