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Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox without a proof construction?

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    I'm voting down as I don't think this question makes sense. As far as I know there's no relationship between Russell's paradox and constructive mathematics, so I don't understand what the question is asking.2010-07-21
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    @Noah: There is, at least historically - Weyl (1921) "On the New foundational Crisis of Mathematics" talks of paradoxes threateneing the chorence of mathematics as an enterprise, and recommending intuitionistic or predicative mathematics as the solution.2010-07-21
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    @Noah, they are related in a strong sense. I suggest you read about the history of constructivism. Intuitionists were trying to find a safer foundations that would avoid paradoxes arising in classical mathematics and set theory like Russell's paradox.2011-11-12
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    In fact the Russell paradox is perfectly constructive. It shows that naïve set theory is inconsistent, and does so in an entirely constructive way. No excluded middle is used, because even intuitionistically an equivalence $P\iff\lnot P$ is a contradiction (where $P$ is $R\in R$ for the set $R=\{x\mid x\notin x\}$): one has $(P\to\lnot P)\implies\lnot P$ and $(\lnot P\to P)\implies\lnot\lnot P$, and of course $\lnot P\land\lnot\lnot P\implies\bot$). The problem is with the axioms of naïve set theory, not with the logic used.2013-03-04

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