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Cumulative HigherOrder Logic as a Foundation for Set Theory
"... The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the socalled 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 ..."
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The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the socalled 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 rstorder set theories can be interpreted in K 1 , for suitable . Keywords: cumulative types, innitary inference rule, logical foundations of set theory. MSC: 03B15 03B30 03E30 03F25 1 Introduction The idea of founding mathematics on a theory of types was rst proposed by Russell [20] (foreshadowed already in [19]), and subsequently implemented by Whitehead and Russell [26]. The formal systems presented in these works were later simplied and cast into their modern shape by Ramsey [18]. Godel [9] and Tarski [25] were the rst to restrict the type structure to types of unary predi...
Cumulative HigherOrder Logic
"... Abstract The systems Kff of transfinite cumulative types up to ff are extended tosystems K1ff that include a natural infinitary inference rule, the socalledlimit rule. For countable ff a semantic completeness theorem for K1ff isproved by the method of reduction trees, and it is shown that every mod ..."
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Abstract The systems Kff of transfinite cumulative types up to ff are extended tosystems K1ff that include a natural infinitary inference rule, the socalledlimit rule. For countable ff a semantic completeness theorem for K1ff isproved by the method of reduction trees, and it is shown that every model of K1ff is equivalent to a cumulative hierarchy of sets. This is used to showthat several axiomatic firstorder set theories can be interpreted in