Abstract

Here it is shown that a certain non-trivial fragment of arithmetic can be interpreted in some propositional calculus. Arithmetical formulae are interpreted there as sets of rules of our calculus. Since among interpretable arithmetical formulae there are some which are as yet unsolved arithmetical problems, we are unable to say about some sets of rules whether they contain, any rules of our calculus or not. I think that there is’a conviction about sufficiency of arithmetical tools for solving problems of propositional calculi. In view of the above remark this conviction is false. Even so we can show some interesting properties of our calculus. Eventually we show that our calculus determines some topological space, so that we can speak about an interpretation of the mentioned fragment of arithmetic in this topology