Philosophical foundations of mathematics and mathematical foundation of metaphysics (some views on improbable god-machines)

Overview & Synopsis

In this article, we first examine the various ways in which set theory has attempted to avoid paradoxes (Cantor, Russell, Burali-Forti, etc.). The classical distinction between "sets" and "proper classes" undoubtedly succeeds in preventing them, but this pure prohibition cannot answer the ontological question of the existence of these classes. Set theories with a universal set (NF, NFU, KF, MK, etc.) unfortunately fare no better. Therefore, associating the existence of a metaphysical entity (a superset, in a sense) with these theories, in the hope of formalizing Anselm of Canterbury's ontological argument and thus providing a mathematical model of "the being of which there cannot be a greater" seems to us a rather futile endeavor. As futile, in truth, as using other encompassing theories (Grothendieck universe, topos), their associated logical programs or computational architectures to give meaning to the idea of an "Absolute Machine." Today, we can only abandon this kind of perspective, without having anything else to offer in its place. Unless we fall into polytheism or consider associating a cascade of simulations with a series of more or less powerful gods (as in Samuel Butler's theory), we can only conclude that we are facing a kind of negative atheism.