He found the incompleteness theorem
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No. If you work within its definitions and the rules of logic it is not flawed. There are mathematical statements that you cannot prove to be true or false (Godel's incompleteness theorem), but that is not a flaw.
Longer than you or anyone else will live! Thanks to Godel, there are statements about mathematical systems such that neither they, not their negation, can ever be proven to be true. This allows a whole new family of mathematical thinking to develop.
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Mathematics"mathematics" is a plural noun already, the subject is Mathematics!
mathematics