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I am not sure what you mean by the number system failing. One possible failure is Godel's incompleteness theorems. According to the first of these, in any consistent system of axioms whose theorems can be listed by an effective procedure is not capable of proving all truths about the arithmetic of the natural numbers. In any such system there will always be statements about the natural numbers that are true, but that are cannot be proved within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

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Q: Why did the whole number system fail?
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