Abstract

If the philosophy of mathematics wants to be rigorous, the concept of infinity must stop being equivocal (both potential and actual) as it currently is. The conception of infinity as actual is responsible for all the paradoxes that compromise the very foundation of mathematics and is also the basis on which Cantor's argument is based on the non-countability of R, and the existence of infinite cardinals of different magnitude. Here we present proof that all infinite sets (in a potential sense) are countable and that there are no infinite cardinals. This article presents a new argument against the existence of the Platonic world of ideas, the ontological basis for the actual infinity. This allows us to deny mathematical Platonism and adopt a non-subjective psychological realism that explains the effectiveness of mathematics in physics and that can go beyond the scope of mathematics.