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## On the cardinality of formula bases and absence of finite approximation with respect to completeness in propositional provability logic. (Russian)

Authors: Rusu Andrei

### Abstract

The probability logic \$G\$ permits to immerse the intuitionistic logic into Peano Arithmetic. Its formulas are built by connectives \$\Cal E\$, \$\vee\$, \$\supset\$, \$\neg\$, \$\bigtriangleup\$. A system \$\Sigma\$ of formulas is called complete (with respect to expressibility) in the logic \$L\$ if all formulas are expressible via \$\Sigma\$ in \$L\$. A system of formulas is reffered to as formula basis in the logic \$L\$, if it is complete in \$L\$, but every proper subsystem of it is incomplete in \$L\$. Theorem 1. For every integer positive \$k\$ there are formula bases of power \$k\$ in the logic \$G\$ and in its extentions, which are not local tabular. A logic \$L\$ is called finitely approximable with respect to completeness, if for every finite incomplete in \$L\$ system \$\Sigma\$ there exists an extension of the logic \$L\$ in which \$\Sigma\$ is incomplete too. Theorem 3. The logic \$G\$ and its extensions, which are not local tabular, are not finitely approximable with respect to completeness. Theorem 5. An extension of logic \$G\$ is finitely approximable with respect to completeness if and only if it is local tabular.