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Proof theory is an area of logic that studies proof as formal mathematical objects. If you'd like advice on the presentation of a proof you have in draft, use proof-writing instead. If you'd like feedback on its validity, use proof-verification. If none of the above apply, you do not need a proof-* tag.

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Are the "proofs by contradiction" weaker than other proofs?

In mathematics you can construct a mathematical theory with different sets of axioms. This can be really useful. When mathematicians ignored the parallel line axiom in Euclidian geometry it gave raise …
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