Gabriel Ciobanu
Alexandru Ioan Cuza University & IIT Romanian Academy, Iaşi, Romania

Mathematics is based heavily on intuition. Regarding the reliability of our intuitions, we decide how credulous or sceptical we are when setting out the fundamental beliefs (based on perception, experience, caution). The axiom of choice (AC) allows us to make an infinite number of arbitrary choices; it has generated a large amount of controversy (Banach-Tarski paradox is one of the non-intuitive consequence of AC). Hilbert's finitism says that any theorem about finite objects that can be obtained using ideal infinite objects can be also obtained without them. We extend this finitism by using the set theory with atoms ZFA (including the axiom of infinity) by adding a new axiom saying that we work with finitely supported sets. Inspired by the permutation models (introduced in 1922 by Fraenkel and developed by Mostowski in 1938 to show the independence of AC from the other axioms of ZF/ZFA), the finitely supported sets are infinite sets having finitely many elements up to permutations of the underlying structure of atoms. Many ZF results can be translated in this new frame by replacing `(non-atomic) set' with `finitely supported (atomic) set'. We prove that most of the important ZF results are valid for finitely supported sets. However, some results are no longer valid in this framework. We prove that AC and the choice principles are not valid in this new set theory based on finitely supported sets. Moreover, the continuum hypothesis is not valid in this extended version of finitism based on finitely supported sets.