Abstract:
We state the following results: the family of all infinite
computably enumerable sets has no computable numbering; the family
of all infinite $\Pi^{1}_{1}$ sets has no
$\Pi^{1}_{1}$-computable numbering; the family
of all infinite
$\Sigma^{1}_{2}$ sets has no
$\Sigma^{1}_{2}$-computable numbering. For $k>2$,
the existence of a
$\Sigma^{1}_{k}$-computable numbering for the family of all
infinite
$\Sigma^{1}_{k}$ sets leads to the inconsistency of $ZF$.
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