Computing sets from all infinite subsets

Noam Greenberg (1) , Matthew Harrison-Trainor (1) , Ludovic Patey (2, 3) , Dan Turetsky (1)

1 School of Mathematics, Statistics and Computer Science [Wellington]
2 CNRS - Centre National de la Recherche Scientifique
3 AGL - Algèbre, géométrie, logique

A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is Π 1 1-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset.

Domaines: Logique [math.LO]

Noam Greenberg, Matthew Harrison-Trainor, Ludovic Patey, Dan Turetsky. Computing sets from all infinite subsets. Transactions of the American Mathematical Society, 2021, 374 (11), pp.8131--8160. ⟨10.1090/tran/8468⟩. ⟨hal-02992972⟩