We consider clopen game formulae, analogous to the open game formulae widely studied in admissible recursion theory. This leads to characterizing the locally countable ${\mathrm{\Sigma }}_1^1$-reflecting ordinals α as those for which, over $L_{\alpha }$, the clopen-game–definable relations are not precisely those which are both open-game and closed-game definable.

中文翻译：

---

一个表征 ${\mathrm{\Sigma }}_1^1$-反映序数

我们考虑封闭博弈公式，类似于在可容许递归理论中广泛研究的开放博弈公式。这导致表征局部可数${\mathrm{\Sigma }}_1^1$-将序数α反映为那些，超过${升}_{\alpha }$，封闭博弈-可定义关系不完全是那些既可定义又可定义开放博弈的关系。