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    Russell's paradox showed that a set R could both be and n... — Carmelics
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    Challenges→The axioms Frege was using to formalize his logic were inconsistent

    Russell's paradox showed that a set R could both be and not be a member of itself within Frege's system

    Philosophy of LanguageTruth & Knowledge
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    Frege's Axiom V allowed an expression φ(x) to be considered both a function of t...The axioms Frege was using to formalize his logic were inconsistentThis ambiguity in Axiom V allowed Russell to construct the paradoxical set R

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    Russell's paradox revealed a contradiction in early set theory84%Russell's paradox demonstrates that unrestricted self-referential set ...83%A set cannot both be and not be a member of itself simultaneously81%The Burali-Forti paradox and other paradoxes further undermined early ...80%

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    Russell wrote to Frege with news of his paradox on June 16, 1902. (For the relevant correspondence, see Russell (1902) and Frege (1902) in van Heijenoort (1967).) The paradox was of significance to Frege’s logical work since, in effect, it showed that the axioms Frege was using to formalize his logic were inconsistent. Specifically, Frege’s Axiom V requires that an expression such as \(\phi(x)\) be considered both a function of the argument \(x\) and a function of the argument \(\phi\). (More pr

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