The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissi...

0. Introduction 1 1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The back-and-forth construction 3 1.3. The Scott isomorphism theorem 4 1.4. ω-logic 7 1.5. Familiar theorems 8 1.6. Failure of compactness 9 2. Background on admissible sets 10 2.1. ∆0 formulas and Σ-formulas in set theory 10 2.2. Axioms of KP 11 2.3. Examples of admissible sets 12 2.4. The admissible set...

Visser’s rules form a basis for the admissible rules of IPC. Here we show that this result can be generalized to arbitrary intermediate logics: Visser’s rules form a basis for the admissible rules of any intermediate logic L for which they are admissible. This implies that if Visser’s rules are derivable for L then L has no non-derivable admissible rules. We also provide a necessary and suffici...

Best-response sets (Pearce [29, 1984]) characterize the epistemic condition of “rationality and common belief of rationality.” When rationality incorporates a weak-dominance (admissibility) requirement, the self-admissible set (SAS) concept (Brandenburger-Friedenberg-Keisler [18, 2008]) characterizes “rationality and common assumption of rationality.” We analyze the behavior of SAS’s in some ga...