University Seminar- Computability, Complexity, and Algebraic Structure--Never-Two Theorem

Time: Friday, February 7, 3:05 – 4:05pm

Place: Phillips Hall, Room 209

Speaker:  Paula de Lima Souza, GWU

Title: Never-Two Theorem

Abstract:  Given any complete theory T, it is natural to ask “How many countable models of T are there, up to isomorphism?” One can easily construct theories that have exactly one, countably many, or continuum many countable models. In the finite case, we may ask for which numbers n there is a theory with exactly n nonisomorphic models. In this talk, we will prove the surprising result proven by Robert Vaught, which states that T can have n many models for any natural number n, except for n=2. In the computable context, one may ask if there is an effective version of Vaught's theorem, that is, how many countable decidable models can a complete decidable theory T have up to isomorphism? In contrast to the previous theorem, we present a result established by Millar and Kudaibergenov that there is a complete decidable theory with exactly two non-isomorphic decidable models.