We distinguish between two kinds of mathematical assertions: objective and constructive. An objective assertion describes the universe of mathematical objects; a constructive one describes the (idealized) mathematician's ability to find mathematical objects with various properties. The familiar formalizations of classical mathematics are based on formal languages designed for expressing objective assertions only. The constructivist program stresses, on the contrary, the importance of constructive assertions; moreover, intuitionism claims that constructive activities of the mind constitute the very subject matter of mathematics, and thus questions the semantic status of objective assertions.
The purpose of this paper is to show that classical mathematics can be extended to include constructive sentences, so that both objective and constructive properties can be discussed in the framework of the same theory. To achieve this goal, we introduce a new property of mathematical objects, calculability.
The word “calculable” may be applied to objects of various types: natural numbers, integers, rational or real numbers, polynomials with rational or real coefficients, etc. In each case it has a different meaning, so that actually we define not one, but many new properties.