2024/11/18 by Gonzalez, David, Harrison-Trainor, Matthew · 1 citation
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2411.12084
We demonstrate that any Πα sentence of the infinitary logic Lω1 ω extending the theory of linear orderings has a model with a Πα+4 Scott sentence and hence of Scott rank at most α+3. In other words, the gap between the complexity of the theory and the complexity of the simplest model is always bounded by 4. This contrasts the situation with general structures where for any α there is a Π2 sentence all of whose models have Scott rank α. We also give new lower bounds, though there remains a small gap between our lower and upper bounds: For most (but not all) α, we construct a Πα sentence extending the theory of linear orderings such that no models have a Σα+2 Scott sentence and hence no models have Scott rank less than or equal to α.