Cumulative Higher-Order Logic as a Foundation for Set Theory

The systems K of transsnite cumulative types up to are extended to systems K 1 that include a natural innnitary inference rule, the so-called limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic rst-order set theories can be interpreted in K 1 , for suitable .dations of set theory.