Hilbert believed that everything in mathematics could and should be proved from the basic axioms. The result of this would be to demonstrate conclusively the two most important elements of the mathematical system. First, mathematics should, at least in theory, be able to answer every single question—this is the same ethos of completeness that had in the past demanded the invention of new numbers like the negatives and the imaginaries. Second, mathematics should be free of inconsistencies—that is to say, having shown that a statement is true by one method, it should not be possible to show that
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