We point out an existence criterion for positive computable total Π 1 1 $$ {\Pi}_1^1 $$ -numberings of families of subsets of a given Π 1 1 $$ {\Pi}_1^1 $$ -set. In particular, it is stated that the family of all Π 1 1 $$ {\Pi}_1^1 $$ -sets has no positive computable total Π 1 1 $$ {\Pi}_1^1 $$ -numberings. Also we obtain a criterion of existence for computable Friedberg Σ 1 1 $$ {\Sigma}_1^1 $$ -numberings of families of subsets of a given Σ 1 1 $$ {\Sigma}_1^1 $$ - set, the consequence of which is the absence of a computable Friedberg Σ 1 1 $$ {\Sigma}_1^1 $$ -numbering of the family of all Σ 1 1 $$ {\Sigma}_1^1 $$ -sets. Questions concerning the existence of negative computable Π 1 1 $$ {\Pi}_1^1 $$ - and Σ 1 1 $$ {\Sigma}_1^1 $$ -numberings of the families mentioned are considered.

中文翻译：

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超算术中的可计算正数和弗里德伯格数

我们指出了正可计算总 Π 1 1 $$ {\Pi}_1^1 $$ -给定 Π 1 1 $$ {\Pi}_1^1 $$ -set 的子集族的存在标准。特别地，指出所有 Π 1 1 $$ {\Pi}_1^1 $$ -sets 的族没有可计算的正总数 Π 1 1 $$ {\Pi}_1^1 $$ -numberings。我们还获得了可计算的 Friedberg Σ 1 1 $$ {\Sigma}_1^1 $$ - 给定 Σ 1 1 $$ {\Sigma}_1^1 $$ - 子集族的存在标准，其结果是缺少可计算的弗里德伯格 Σ 1 1 $$ {\Sigma}_1^1 $$ -所有 Σ 1 1 $$ {\Sigma}_1^1 $$ - 集合的族的编号。考虑了关于所提到的家庭的负可计算 Π 1 1 $$ {\Pi}_1^1 $$ - 和 Σ 1 1 $$ {\Sigma}_1^1 $$ - 编号的存在性问题。