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The consistency problem for positive comprehension principles. (English) Zbl 0702.03026

The paper extends the results by P. Gilmore [Axiomatic set theory, Proc. Symp. Los Angeles 1967, 147-153 (1974; Zbl 0309.02065)] on consistency of unrestricted comprehension \(\exists a\forall x(x\in a\leftrightarrow \phi)\) for positive formulas \(\phi\). In addition to the language \({\mathcal L}=\{\in,=\}\) the language \({\mathcal L}_{{\mathcal T}}\) allowing abstracts \(\{\) \(t|\phi\) (t)\(\}\) is considered. Positive comprehension (p.c.) for \({\mathcal L}_{{\mathcal T}}\) is the schema \(\forall x(x\in \{t|\phi\) (t)\(\}\leftrightarrow \phi (x))\) with formulas \(\phi\) containing negation only inside abstracts. EXT is the extensionality axiom. Gilmore’s construction shows that p.c. for \({\mathcal L}_{{\mathcal T}}+EXT+\exists xy(x\neq y)\) is inconsistent. Without extensionality p.c. for \({\mathcal L}_{{\mathcal T}}\) plus infinity axiom has a model in Zermelo set theory with separation schema restricted to \(\Delta_ 0\)-formulas. If quantifiers relativized to arbitrary formulas are allowed, the consistency of p.c. generalized in this way plus infinity axiom follows from the consistency of ZF. This is obtained by adaptation of a construction of Malitz. In Quine’s New Foundations (extended if necessary by \(\exists xy(x\neq y))\) it is possible to take p.c. in \({\mathcal L}_{{\mathcal T}}\) for stratified formulas as the only comprehension scheme.
Reviewer: G.Mints

MSC:

03E35 Consistency and independence results

Citations:

Zbl 0309.02065
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