Submission date: 13 September 2021

Abstract:

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence Φ as a class of structures in a related language. From this, we show that Φ has a Borel complete expansion if and only if S_∞ divides Aut(M) for some countable model M ⊨ Φ. Using this, we prove that for theories T_h asserting that {E_n} is a countable family of cross cutting equivalence relations with h(n) classes, if h(n) is uniformly bounded then T_h is not Borel complete, providing a converse to Theorem 2.1 of [LU].

Mathematics Subject Classification: 03C55 03E15

Keywords and phrases:

Full text arXiv 2109.06140: pdf, ps.