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On the Consistency of a Positive Theory

โœ Scribed by Olivier Esser

Publisher
John Wiley and Sons
Year
1999
Tongue
English
Weight
735 KB
Volume
45
Category
Article
ISSN
0044-3050

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โœฆ Synopsis

Abstract

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the โ€œgeneralized positiveโ€ theory GPK~โˆž~^+^. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK~โˆž~^+^ interprets the Kelley Morse class theory. Here we prove that GPK~โˆž~^+^ + AC~WF~ (AC~WF~ being a form of the axiom of choice allowing to choose elements in wellโ€founded sets) and the Kelleyโ€Morse class theory with the axiom of global choice and the axiom โ€œOn is ramifiableโ€ are mutually interpretable. This shows that GPK~โˆž~^+^ + AC~WF~ is a โ€œstrongโ€ theory since โ€œOn is ramifiableโ€ implies the existence of a proper class of inaccessible cardinals.

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