Complexity of the axioms of the alternative set theory

Antonín Sochor

Commentationes Mathematicae Universitatis Carolinae (1993)

  • Volume: 34, Issue: 1, page 33-45
  • ISSN: 0010-2628

Abstract

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If T is a complete theory stronger than ZF Fin such that axiom of extensionality for classes + T + ( ∃ X ) Φ i is consistent for 1 ≤ i ≤ k (each alone), where Φ i are normal formulae then we show AST + ( ∃ X ) Φ 1 + ⋯ + ( ∃ X ) Φ k + scheme of choice is consistent. As a consequence we get: there is no proper Δ 1 -formula in AST + scheme of choice. Moreover the complexity of the axioms of AST is studied, e.gẇe show axiom of extensionality is Π 1 -formula, but not Σ 1 -formula and furthermore prolongation axiom, axioms of choice and cardinalities are Π 2 -formulae, but not Π 1 -formulae in AST without the axiom in question.

How to cite

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Sochor, Antonín. "Complexity of the axioms of the alternative set theory." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 33-45. <http://eudml.org/doc/247496>.

@article{Sochor1993,
abstract = {If T is a complete theory stronger than ZF$_\{\hbox\{Fin\}\}$ such that axiom of extensionality for classes + T + $(\exists X)\Phi _i$ is consistent for 1$\le i \le k$ (each alone), where $\Phi _i$ are normal formulae then we show 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 AST + scheme of choice. Moreover the complexity of the axioms of AST is studied, e.gẇe 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 AST without the axiom in question.},
author = {Sochor, Antonín},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {alternative set theory; complexity of formulae; $\Pi _2$-formula; extension of axiomatic systems; complexity of formulas; -formula; extension of axiomatic system; prolongation; alternative set theory; choice; cardinalities},
language = {eng},
number = {1},
pages = {33-45},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Complexity of the axioms of the alternative set theory},
url = {http://eudml.org/doc/247496},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Sochor, Antonín
TI - Complexity of the axioms of the alternative set theory
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 33
EP - 45
AB - If T is a complete theory stronger than ZF$_{\hbox{Fin}}$ such that axiom of extensionality for classes + T + $(\exists X)\Phi _i$ is consistent for 1$\le i \le k$ (each alone), where $\Phi _i$ are normal formulae then we show 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 AST + scheme of choice. Moreover the complexity of the axioms of AST is studied, e.gẇe 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 AST without the axiom in question.
LA - eng
KW - alternative set theory; complexity of formulae; $\Pi _2$-formula; extension of axiomatic systems; complexity of formulas; -formula; extension of axiomatic system; prolongation; alternative set theory; choice; cardinalities
UR - http://eudml.org/doc/247496
ER -

References

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  10. Sochor A., Vopěnka P., Shiftings of the horizon, Comment. Math. Univ. Carolinae 24 (1983), 127-136. (1983) MR0703932
  11. Sgall J., Construction of the class FN, Comment. Math. Univ. Carolinae 27 (1986), 435-436. (1986) Zbl0611.03025MR0873617
  12. 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). 
  13. Vopěnka P., Mathematics in the Alternative Set Theory, TEUBNER TEXTE, Leipzig, 1979. MR0581368
  14. Vopěnka P., Úvod do matematiky v alternatívnej teórii množín (in Slovak), ALFA Bratislava, 1989. 

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