Activity: Talk or presentation › Invited talk › Popular
Description
Sequential Theories form a natural class of theories that include set theories and arithmetical theories. Examples of sequential theories are Gödel-Bernays Set Theory, Zermelo-Fraenkel Set Theory, Peano Arithmetic, I\Sigma_1, I\Delta_0, S^1_2, PA^-. Typical for sequential theories is the possibility to develop partial satisfaction predicates for the full language of the given theory. As a consequence, sequential theories are locally essentially reflexive: they prove
restricted reflection principles at the cost of choosing deeper definable cuts for stronger reflection principles.
In our talk we survey the state of the art concerning sequential theories. We discuss the behavior of these theories w.r.t. mutual interpretability. We explain the Friedman Characterization of interpretability among finitely axiomatized sequential theories and the analogue of the Orey-Hájek Characterization in the infinitely axiomatized case. We describe Friedman's Theorem that interpretations between restricted interpretations can always be replaced by faithful ones.
We provide examples of interesting properties of the degrees of interpretability of sequential theories. Finally, we treat some model theory of sequential theories.