玄妙基数

超限数を扱う数学において、玄妙基数（げんみょうきすう、英: Ineffable cardinal）は巨大基数の一種でJensen & Kunen (1969)によって導入された。

基数 $\kappa$ がほとんど玄妙であるとは、全ての順序数 $\delta <\kappa$ に対して $f(\delta )$ が $\delta$ の部分集合となるような全ての関数 $f:\kappa \to {\mathcal {P}}(\kappa )$ (ここで ${\mathcal {P}}(\kappa )$ は $\kappa$ の冪集合)に対して、 $\kappa$ のある濃度 $\kappa$ の部分集合 $S$ が存在して、 $f$ に対してhomogeneous（すなわち、 $S$ の要素である全ての $\delta _{1}<\delta _{2}$ に対して、 $f(\delta _{1})=f(\delta _{2})\cap \delta _{1}$ ）となること。

基数 $\kappa$ が玄妙であるとは、全ての2値関数 $f:{\mathcal {P}}_{=2}(\kappa )\to \{0,1\}$ に対し $f$ に対してhomogeneousとなる $\kappa$ の定常集合があること。: すなわち、 $f$ がその定常集合の要素である非順序対を全て0に送るか、全て1に送ること。

もっと一般的に $\kappa$ が $n$ -玄妙(ただし $n$ は正の整数) とは、全ての関数 $f:{\mathcal {P}}_{=n}(\kappa )\to \{0,1\}$ に対して $f$ に対して $n$ -homogeneous となる $\kappa$ の定常集合が存在する (すなわち、その定常集合の要素である $n$ -非順序対を全て同じ値に送る)こと。すなわち、玄妙は2-玄妙と同じ意味である。

完全玄妙基数とは、 $2\leq n<\aleph _{0}$ となる全ての n に対して $n$ -玄妙となる基数のこと。 $\kappa$ が $(n+1)$ -玄妙であれば $\kappa$ 以下の $n$ -玄妙基数の集合は $\kappa$ の定常部分集合となる。

完全玄妙基数は精妙基数(subtle cardinal)より強い無矛盾性を持ち、 remarkable cardinalより弱い無矛盾性を持つ。巨大基数公理の無矛盾性の強さの表はここにまとめられている。

参照

Friedman, Harvey (2001), “Subtle cardinals and linear orderings”, Annals of Pure and Applied Logic 107 (1–3): 1–34, doi:10.1016/S0168-0072(00)00019-1 .
Jensen, R. B.; Kunen, K. (1969), Some Combinatorial Properties of L and V, Unpublished manuscript

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