Complexity of the axioms of the alternative set theory
Sochor, Antonín
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993), p. 33-45 / Harvested from Czech Digital Mathematics Library
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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.

Publié le : 1993-01-01
Classification:  03A05,  03D55,  03E30,  03E35,  03E70,  03H05,  03H15 
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     author = {Anton\'\i n Sochor},
     title = {Complexity of the axioms of the alternative set theory},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {34},
     year = {1993},
     pages = {33-45},
     zbl = {0792.03037},
     mrnumber = {1240201},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118553}
}

Sochor, Antonín. Complexity of the axioms of the alternative set theory. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 33-45. http://gdmltest.u-ga.fr/item/118553/

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