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Even though the notion of infinity has been proven useful in a lot of analysis methods today I will argue that sometimes it creates fundamental problems that could have been avoided. Let's start with one of the pillars of modern Computer Science - Turing's Halting problem. If we do not assume the existence of infinite-tape infinite-state Turing Machines it becomes clear that for all Turing Machines with at most N states and the size of the tape of at most M we can construct a Halting Problem sol...