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Inconsistency of the Axiom of Choice with the Positive Theory $GPK^+ \infty$

Journal of Symbolic Logic 65 (4):1911-1916 (2000)

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Choice principles in hyperuniverses.Marco Forti & Furio Honsell - 1996 - Annals of Pure and Applied Logic 77 (1):35-52.details It is well known that the validity of Choice Principles is problematic in non-standard Set Theories which do not abide by the Limitation of Size Principle. In this paper we discuss the consistency of various Choice Principles with respect to the Generalized Positive Comprehension Principle . The Principle GPC allows to take as sets those classes which can be specified by Generalized Positive Formulae, e.g. the universe. In particular we give a complete characterization of which choice principles hold in Hyperuniverses. (...) Download Export citation Bookmark 8 citations
Inconsistency of GPK + AFA.Olivier Esser - 1996 - Mathematical Logic Quarterly 42 (1):104-108.details M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X1 introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X1 in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It (...) Download Export citation Bookmark 4 citations
An Interpretation of the Zermelo‐Fraenkel Set Theory and the Kelley‐Morse Set Theory in a Positive Theory.Olivier Esser - 1997 - Mathematical Logic Quarterly 43 (3):369-377.details An interesting positive theory is the GPK theory. The models of this theory include all hyperuniverses (see [5] for a definition of these ones). Here we add a form of the axiom of infinity and a new scheme to obtain GPK∞+. We show that in these conditions, we can interprete the Kelley‐Morse theory (KM) in GPK∞+ (Theorem 3.7). This needs a preliminary property which give an interpretation of the Zermelo‐Fraenkel set theory (ZF) in GPK∞+. We also see what happens in (...) Download Export citation Bookmark 7 citations
Addendum and corrigendum Choice Principles in Hyperuniverses Annals of Pure and Applied Logic 77 (1996) 35–52.Marco Forti & Furio Honsell - 1998 - Annals of Pure and Applied Logic 92 (2):211-214.details The proof of Lemma 5 in our paper “Choice Principles in Hyperuniverses” [3], contains an error. In the present note we show that the statement of that lemma is false and hence the Axiom of Choice fails in all κ-hyperuniverses, for uncountable κ. However, a weaker version of Lemma 5 can be proved, which implies that the Linear Ordering Principle holds in all κ-metric κ-hyperuniverses. Download Export citation Bookmark 2 citations
On the Consistency of a Positive Theory.Olivier Esser - 1999 - Mathematical Logic Quarterly 45 (1):105-116.details In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK∞+. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK∞+ interprets the Kelley Morse class theory. Here we prove that GPK∞+ + ACWF and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK∞+ + ACWF is a “strong” theory since “On (...) Download Export citation Bookmark 14 citations

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