On abstract model theory and defining well-orderings

Abstract

In this paper we will study the expressive power, measured by the ability to define certain classes, of some extensions of first order logic. The central concepts will be definability of classes of ordinals and the well-ordering number w of a logic. First we discuss the partial orders ≤, ≤P C and ≤RP C on logics and how these relate to each other and to our definability concept. Then we study the division between bounded and unbounded logics. An interrest- ing result in this direction is the theorem due to Lopez-Escobar stating that L∞ω is weak in the sense that it does not define the entire class of well-orderings, even though it has no well-ordering number, whereas Lω1 ω1 is strong in the same sense.

In this paper we will study the expressive power, measured by the ability to define certain classes, of some extensions of first order logic. The central concepts will be definability of classes of ordinals and the well-ordering number w of a logic. First we discuss the partial orders ≤, ≤P C and ≤RP C on logics and how these relate to each other and to our definability concept. Then we study the division between bounded and unbounded logics. An interrest- ing result in this direction is the theorem due to Lopez-Escobar stating that L∞ω is weak in the sense that it does not define the entire class of well-orderings, even though it has no well-ordering number, whereas Lω1 ω1 is strong in the same sense.