A.

In the paper "SCIENTIFIC INTUITION OF GENII AGAINST MYTHO-'LOGIC' OF CANTOR'S TRANSFINITE 'PARADISE'" by A.A.Zenkin (presented at the International Symposium "Philosophical Insights into Logic and Mathematics", Nancy, France, 30September-04October, 2002), new aspects of Cantor's diagonal proof of the uncountability of continuum are presented and analyzed.

Abstract. - It is shown that the famous Cantor's theorem on the uncountability of continuum was never comprising at least two necessary conditions of its own proof in an explicit form. The scandal, from the mathematical point of view, fact explains, in particular, why nobody was able to disprove the theorem. The explication of the conditions and their logical analysis prove that Cantor's "diagonal argument" proves nothing: the first necessary condition makes the "diagonal argument" invalid, the second necessary condition is simply a teleological one having no relation to mathematics. Thus, the notorious uncountability of continuum is not proved by Cantor, and our results show that the main paradigms of foundations of modern meta-mathematics (so-called 'proof theory') and 'non-naive' axiomatic set theory must be essentially revised from the point of view of really working, classical mathematics.