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

Cumulative Higher-Order Logic as a Foundation for Set Theory