On the Basics of Mathematics

As shown in the works of Gödel and others, standard mathematics has foundational problems because it invokes the infinite ring of integers Z. At the same time, finite mathematics proceeds from a finite ring Rp=(0,1,2,...p-1) of residues modulo p. We give a new and simple proof that Z is the limit of Rp when \( p\to\infty \). The proof involves only potential infinity and not actual infinity. We explain why the proof plays an important role in the foundations of mathematics and physics.