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Title: Complexity of the axioms of the alternative set theory (English)
Author: Sochor, A.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 34
Issue: 1
Year: 1993
Pages: 33-45
Category: math
Summary: If {\bf T} is a complete theory stronger than {\bf ZF}$_{\hbox {Fin}}$ such that axiom of extensionality for classes + {\bf T} + $(\exists X)\Phi_i$ is consistent for 1$\leq i \leq k$ (each alone), where $\Phi_i$ are normal formulae then we show {\bf AST} + $(\exists X)\Phi_1 +\dots + (\exists X)\Phi_k$ + scheme of choice is consistent. As a consequence we get: there is no proper $\Delta_1$-formula in {\bf AST} + scheme of choice. Moreover the complexity of the axioms of {\bf AST} is studied, e.g\. we show axiom of extensionality is $\Pi_1$-formula, but not $\Sigma_1$-formula and furthermore prolongation axiom, axioms of choice and cardinalities are $\Pi_2$-formulae, but not $\Pi_1$-formulae in {\bf AST} without the axiom in question. (English)
Keyword: alternative set theory
Keyword: complexity of formulae
Keyword: $\Pi_2$-formula
Keyword: extension of axiomatic systems
MSC: 03A05
MSC: 03D55
MSC: 03E30
MSC: 03E35
MSC: 03E70
MSC: 03H05
MSC: 03H15
idZBL: Zbl 0792.03037
idMR: MR1240201
Date available: 2009-01-08T18:00:53Z
Last updated: 2012-04-30
Stable URL:
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