University Seminars:Logic Across Disciplines-Scott Ranks of Scattered Linear Orders

Title:Scott Ranks of Scattered Linear Orders
Speaker: Rachael Alvir, University of Notre Dame
Date and Time:  Thursday, April 26, 2018 12:30PM-1::30 PM
Place: Rome Hall(801 22nd Street), Room 351

Abstract: The logic L(omega1, omega) is obtained from regular finitary first-order logic by closing under countable conjunctions and disjunctions. There is a kind of normal form for such sentences. The Scott rank of a countable structure A is the least complexity of a sentence A of L(omega1, omega), which describes A up to isomorphism among countable structures. Every scattered linear order is associated with an ordinal known as its Hausdorff rank. We give sharp upper bounds on the Scott rank of a scattered linear order given its Hausdorff rank, along the way calculating some of the back-and-forth relations on this class. These results generalize previously obtained results on the Scott ranks of ordinals and Hausdorff rank 1 linear orders.