Characterizing the existence of a Borel complete expansion

2021/09/13 by Laskowski, Michael C., Ulrich, Douglas S.
#03C55 03E15 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2109.06140

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\models 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 \citeLU.

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