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Countably many      ℵ  1 {\aleph _1}      -recursively enumerable sets are constructed from which all the      ℵ  1 {\aleph _1}      -recursively enumerable sets can be generated by using countable union and countable intersection. This implies under      V = L V = L       that there exists as well a countable basis for      ∑  n 1 \sum _n^1       sets of reals,      n ⩾  2 n \geqslant 2      . Further under      V = L V = L       the lattice      E ∗  ( ℵ  1 ) \mathcal {E}*({\aleph _1})       of      ℵ  1 {\aleph _1}      -recursively enumerable sets modulo countable sets has only      ℵ  1 {\aleph _1}       many automorphisms.

A countable basis for Σ¹₂ sets and recursion theory on ℵ₁