(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 2, PAGES 417-435

## A two-sorted theory of classes and sets, admitting sets of propositional formulas

V. K. Zakharov
A. V. Mikhalev

Abstract

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 The crisis arisen in the naive set theory in the beginning of the 20th century brought to the origin of such strict axiomatic theories as the \emph{theory of sets in Zermelo--Fraenkel's axiomatics} (ZF) and the \emph{theory of classes and sets in Neumann--Bernays--G\"odel's axiomatics} (NBG). However, in the same time as the naive set theory admitted considering sets of arbitrary objects, such a natural notion as a \emph{set of propositional formulas} became inadmissible in ZF and NBG. In connection with this circumstance some methods of associated admission were developed, the most known of which is the \emph{method of G\"odel's enumeration}.This paper is devoted to a solution of the \emph{full rights admission problem}. An axiomatics of the \emph{two-sorted theory of classes and sets} is exposed in it, which allows to consider sets of propositional formulas equally with sets of object elements. 

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