A countable basis for $\Sigma ^{1}_{2}$ sets and recursion theory on $\aleph _{1}$

Abstract:

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