Geodesic
Complexity of the axioms of the alternative set theory
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 1, pp. 33-45.

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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.
Classification : 03A05, 03D55, 03E30, 03E35, 03E70, 03H05, 03H15
Keywords: alternative set theory; complexity of formulae; $\Pi_2$-formula; extension of axiomatic systems 
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Sochor, A. Complexity of the axioms of the alternative set theory. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 1, pp. 33-45. http://geodesic.mathdoc.fr/item/CMUC_1993__34_1_a3/