# Article

Full entry | PDF   (0.2 MB)
Keywords:
alternative set theory; complexity of formulae; $\Pi_2$-formula; extension of axiomatic systems
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.
References:
[B] Bernays P.: A system of axiomatic set theory. JSL 2 (1937), 65-77. Zbl 0019.29403
[G] Gödel KI.: The consistency of the axiom of choice and of the general continuum hypothesis. Ann. of Math. Studies, Princeton, 1940.
[P-S1984] Pudlák P., Sochor A.: Models of the alternative set theory. JSL 49 (1984), 570-585. MR 0745386
[S1979] Sochor A.: Metamathematics of the alternative set theory I. Comment. Math. Univ. Carolinae 20 (1979), 697-722. MR 0555184 | Zbl 0433.03028
[S1982] Sochor A.: Metamathematics of the alternative set theory II. Comment. Math. Univ. Carolinae 23 (1982), 55-79. MR 0653351 | Zbl 0493.03030
[S1983] Sochor A.: Metamathematics of the alternative set theory III. Comment. Math. Univ. Carolinae 24 (1983), 137-154. MR 0703933 | Zbl 0531.03031
[S1985] Sochor A.: Constructibility and shiftings of view. Comment. Math. Univ. Carolinae 26 (1985), 477-498. MR 0817822 | Zbl 0583.03040
[S-V1980] Sochor A., Vopěnka P.: Revealments. Comment. Math. Univ. Carolinae 21 (1980), 97-118. MR 0566243
[S-V1981] Sochor A., Vopěnka P.: The axiom of reflection. Comment. Math. Univ. Carolinae 22 (1981), 87-111. MR 0609938
[S-V1983] Sochor A., Vopěnka P.: Shiftings of the horizon. Comment. Math. Univ. Carolinae 24 (1983), 127-136. MR 0703932
[Sg1986] Sgall J.: Construction of the class FN. Comment. Math. Univ. Carolinae 27 (1986), 435-436. MR 0873617 | Zbl 0611.03025
[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).
[V1979] Vopěnka P.: Mathematics in the Alternative Set Theory. TEUBNER TEXTE, Leipzig, 1979. MR 0581368
[V1989] Vopěnka P.: Úvod do matematiky v alternatívnej teórii množín (in Slovak). ALFA Bratislava, 1989.

Partner of