# Complexity of the axioms of the alternative set theory

Commentationes Mathematicae Universitatis Carolinae (1993)

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

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topSochor, 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 -

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