Cardinality logics. I: Inclusions between languages based on ``exactly'' (Q1107521)

In this paper the author develops a logic which includes the notion of cardinal. To do this the author uses different kinds of variables. Variables of type 0 are reserved for elements, variables of type 1 are used for sets (in case they are allowed, then a generalization of second order logic is obtained), variables of type 2 are cardinals. Variables of the next type are used to speak about ``the cardinality of the set of all cardinals less than a given cardinal''. This process is iterated. Cardinals are Scott-cardinals and the author avoids reliance on the Axiom of Choice. The author investigates the expressive power of his languages and introduced severals kinds of hierarchies of languages. A main tool are Ehrenfeucht-games. Some questions remain open. The operator ``exactly'' is used to express that something has a given cardinality.