Authors: Felix M. Lev
It is argued that the main reason of crisis in quantum theory is that nature, which is fundamentally discrete, is described by continuous mathematics. Moreover, no ultimate physical theory can be based on continuous mathematics because, as follows from G\"{o}del's incompleteness theorems, any mathematics involving the set of all natural numbers has its own foundational problems which cannot be resolved. In the first part of the paper inconsistencies in standard approach to quantum theory are discussed and the theory is reformulated such that it can be naturally generalized to a formulation based on discrete and finite mathematics. Then the cosmological acceleration and gravity can be treated simply as {\it kinematical} manifestations of de Sitter symmetry on quantum level ({\it i.e. for describing those phenomena the notions of dark energy, space-time background and gravitational interaction are not needed}). In the second part of the paper motivation, ideas and main results of a quantum theory over a Galois field (GFQT) are described. In contrast to standard quantum theory, GFQT is based on a solid mathematics and therefore can be treated as a candidate for ultimate quantum theory. The presentation is non-technical and should be understandable by a wide audience of physicists and mathematicians.
Comments: 20 Pages.
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