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Do we sometimes prove things based on the assumption that mathematics is self-consistent?

I recently started to be dubious about proofs by contradiction. It seems to me that it is somehow based on the assumption that whatever we do in mathematics we will never arrive at a inconsistency, i.e. something that is proved to be both true and wrong.

Should we worry about this? Are we confident that mathematics is self-consistent? Or is it an issue at all?

Paul
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  • I believe that we do not know if the set of axioms that we typically use for mathematics is self-consistent (the word typically here is somewhat challenging, it depends on how many and which axioms you're using). It is likely unprovable via Godel's incompleteness type theorems to prove consistency. – Michael Burr Mar 06 '16 at 20:21
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    We have no certainty about the consistency of math, but of course we are working "on the assumption that it is consistent". If math is not consistent, then any math sentence is provable... thus, we can simply stop proving theorems. – Mauro ALLEGRANZA Mar 06 '16 at 20:30
  • @MauroALLEGRANZA That's my point. Isn't that a big issue? Not knowing if one day a guy will arrive at a perfectly valid proof that $1=0$ and then everything just collapse. – Paul Mar 06 '16 at 21:04
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    Math has already found contradicitons (see: Russell's paradox) and nothing has "collapsed": the relevant theories have been fixed (and the bridges still stay there, even if they have been "calculated" with inconsistent theories). – Mauro ALLEGRANZA Mar 07 '16 at 07:37

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