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Galileo Galilei famously described our Universe as a grand book written in the language of Mathematics. If our universe is indeed described by mathematics, then the fascinating issues surrounding Godel's theorem are important not only for mathematics, but for physics as well. The goal of the proposed project is to investigate to what extent our successful physical theories like quantum field theory and general relativity require their standard mathematical formulation (using number theory, the uncountable continuum etc.) that leads to Godel undecidability and Church-Turing uncomputability, and to what extent they can be reformulated with finite or at least decidable/computable mathematical structures without violating observational constraints (and if so, what these mathematical differences predict for observational signatures that could be measured in the future). The main activities of the project are research and calculations by Max Tegmark. The main outcome is anticipated to consist of high-impact publications and conference talks, and in the pursuit of an interesting new line of inquiry on incompleteness, at the interface between mathematics and physics.