# Article

 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: http://hdl.handle.net/10338.dmlcz/118553 . Reference: [B] Bernays P.: A system of axiomatic set theory.JSL 2 (1937), 65-77. Zbl 0019.29403 Reference: [G] Gödel KI.: The consistency of the axiom of choice and of the general continuum hypothesis.Ann. of Math. Studies, Princeton, 1940. Reference: [P-S1984] Pudlák P., Sochor A.: Models of the alternative set theory.JSL 49 (1984), 570-585. MR 0745386 Reference: [S1979] Sochor A.: Metamathematics of the alternative set theory I.Comment. Math. Univ. Carolinae 20 (1979), 697-722. Zbl 0433.03028, MR 0555184 Reference: [S1982] Sochor A.: Metamathematics of the alternative set theory II.Comment. Math. Univ. Carolinae 23 (1982), 55-79. Zbl 0493.03030, MR 0653351 Reference: [S1983] Sochor A.: Metamathematics of the alternative set theory III.Comment. Math. Univ. Carolinae 24 (1983), 137-154. Zbl 0531.03031, MR 0703933 Reference: [S1985] Sochor A.: Constructibility and shiftings of view.Comment. Math. Univ. Carolinae 26 (1985), 477-498. Zbl 0583.03040, MR 0817822 Reference: [S-V1980] Sochor A., Vopěnka P.: Revealments.Comment. Math. Univ. Carolinae 21 (1980), 97-118. MR 0566243 Reference: [S-V1981] Sochor A., Vopěnka P.: The axiom of reflection.Comment. Math. Univ. Carolinae 22 (1981), 87-111. MR 0609938 Reference: [S-V1983] Sochor A., Vopěnka P.: Shiftings of the horizon.Comment. Math. Univ. Carolinae 24 (1983), 127-136. MR 0703932 Reference: [Sg1986] Sgall J.: Construction of the class FN.Comment. Math. Univ. Carolinae 27 (1986), 435-436. Zbl 0611.03025, MR 0873617 Reference: [Ve1984] Vencovská A.: Independence of the axiom of choice in the alternative set theory.Open days in model theory and set theory, Proceedings 1981 Jadwisin (Leeds 1984). Reference: [V1979] Vopěnka P.: Mathematics in the Alternative Set Theory.TEUBNER TEXTE, Leipzig, 1979. MR 0581368 Reference: [V1989] Vopěnka P.: Úvod do matematiky v alternatívnej teórii množín (in Slovak).ALFA Bratislava, 1989. .

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