To What Extent Contemporary Mathematical Science is Reliable

The crisis in foundation of mathematics at the end of 19th beginning of 20th centuries initiated a number of axiomatic set theoretical systems during the first half of the 20th century. These systems were the result of different philosophical approaches (in view of second Godel's Theorem) aimed at overcoming of the above crisis. But the way out of this situation has never been found.
In my article I offer a new approach to solve this problem using a basic axiomatic system of the set theory with intuitionistic logic. I will present a lot of mathematical results having been obtained during the last forty years. We will survey the development of the set theory with the intuitionistic logic underlining the main points and formulating unsolved problems and describe the basic system of the intuitionistic set theory.

arithmetic, mathematical analysis, axiomatic systems of set theory, intuitionism, constructivism, basic system, mathematics