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Weight of the comprehension axiom in a theory based on logic without contractions

Authors:

V N Grishin

Institution:

(1) V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Russian

Abstract:

A set-theoretic LST system based on a logic without rules of contraction of identical formulas in the antecedent or succedent of sequents is introduced. The set-theoretic axioms used are the comprehension principle,

$\exists y\forall x\left( {x \in y \equiv A\left( x \right)} \right),$

, where the weight of the variablex in the formulaA(x) is at most one (see below), and the extensionality principle,

$x \subseteq y \wedge y \subseteq x \supset \left( {x \in z \supset y \in z} \right).$

. It is proved that the restriction on the weight of the variablex in the comprehension axiom is essential. Examples of comprehension axioms with the weight of variablex equal to 2 whose combination with the extensionality principle leads to a contradiction in the logic without contraction rules are constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 643–652, November, 1999.

Keywords:

comprehension principle logic without contractions weight of variables comprehension axiom Gentzen calculus of sequents Takahashi rules Girard’ s linear logic

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