CANTOR'S ARITHMETIC OF TRANFINITE NUMBERS, READ TODAY Mathematics is freedom, Physics and Philosophy are not

The historical value of Cantor's work, and also of set theory, both in its naive and axiomatic phases, are undeniable. This article, however, carries out a critical reading of Cantorian theory and, to a large extent, of axiomatic set theory, and connects the analysis with some hints for ongoing discussion in mathematics, physics and epistemology. Inconsistencies and contradictions, in key points of the theory, are highlighted: actual infinity, equal treatment, for most aspects, of infinite and finite sets, infinite plurality and hierarchy of uncountable sets, faultiness of diagonal method. Some of these critical issues have been incorporated, with differences between various versions, into axiomatic theory, which failed to complete the program of providing secure and well-defined foundations for mathematics, which, however, does not appear to suffer from poor health. Mathematicians did not care much about failure of various foundationalist schools, they continued, and it was the best choice, to dedicate themselves to mathematical practice. With the crisis of foundations and with concomitant physical revolutions, the loss of many certainties has occurred, however, the research perspective ends up reversing into an extraordinary open horizon, in mathematics and in borderlands between logic, physics, philosophy and mathematics.