Problems with foundation of mathematics and quantum theory

As shown in Gödel’s incompleteness theorems, standard mathematics has foundational problems because it invokes actual infinity from the very beginning. We describe a theorem which states that the infinite ring of integers Z is a limit of finite rings R_p=(0,1,2,...p-1) of residues modulo p when p→∞, and the proof invokes only potential infinity. Then it can be shown that a finite quantum theory (FQT) in which the state space is a linear space over R_p+iR_p, and operators of physical quantities act in this space, is more general than standard quantum theory (SQT). This means that any result of SQT can be recovered in FQT if p is chosen to be sufficiently large, and at the same time, in FQT there are results which cannot be recovered in SQT, which contains particles and antiparticles (e.g., electron–positron, proton–antiproton) and the laws of conservation of electric charge, baryon number, etc. These results arise because the algebras in SQT are such that in their irreducible representations (IRs), the energies of particles can only be either positive or negative, and there are no IRs in which particles have states with different energy signs. On the contrary, in finite mathematics, IRs necessarily contain both positive and negative energies. The case when there is one IR uniting positive and negative energies has a higher symmetry than the case when this IR splits into two independent IRs with positive and negative energies. Then the concept of particle–antiparticle can only be approximate when p is very large. Therefore, the challenge of constructing fundamental quantum theory lies in constructing a theory that does not assume that p is anomalously large, and such a theory will be based on new physical concepts.