Logic Seminar

The talk is a meld of two papers, both joint with Douglas Ulrich.
We characterize which Φ∈Lω1,ω have Borel complete expansions and give several sufficient conditions for a theory to have a Borel complete reduct, e.g., having uncountably many complete 1-types. We illustrate these general behaviors by considering a family Th of first-order theories, indexed by functions h:ω→ω∖{0,1} in a language L={En:n∈ω} asserting that each En is an equivalence relation with h(n) classes, and that the classes cross-cut. The key to showing that many theories have Borel complete reducts follows from the fact that Th is Borel complete whenever h is unbounded. We deduce our equivalents of having a Borel complete expansion in order to show the converse: if h is bounded, then Th is not Borel complete.