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A consistency result of the system Z \(+\) the replacement axiom schema of \(\Sigma_ n\)-formulas. (English) Zbl 0712.03043

Using the definability of truth for \(\Sigma_ n\)-formulas and the reflection principle the author proves in ZF the consistency of the subtheory \(ZF_ n\) mentioned in the title of his short note. In fact, the author’s (somewhat stronger) result and his proof are well-known and can be found as Theorem 32 in A. Levy’s “A hierarchy of formulas in set theory” [Mem. Am. Math. Soc. 57 (1965; Zbl 0202.305)] (a paper cited by the author in his list of references). Alternatively, one could use for a simple proof the fact that \(ZF_ n\) is finitely axiomatizable [cf. E.-J. Thiele: “Über endlich-axiomatisierbare Teilsysteme der Zermelo-Fraenkelschen Mengenlehre”, Z. Math. Logik Grundlagen Math. 14, 39-58 (1968; Zbl 0177.014)].
Reviewer: K.Gloede

MSC:

03E35 Consistency and independence results
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