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    The paradoxes of set theory are resolved by reducing asse... — Carmelics
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    The paradoxes of set theory are resolved by reducing assertions about sets to assertions about propositional functions.

    Proof of definition segmentsTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Assertions about sets can be reduced to assertions about propositional functions.
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    • 2.The restriction that a function of one type cannot apply to a function of the same type blocks the paradoxes.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Zermelo's axiomatic set theory resolves the same paradoxes without eliminating sets or invoking propositional functions.
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    • 2.When an alternative framework resolves paradoxes with weaker ontological revision, the stronger revision requires additional justification beyond mere paradox-resolution.
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    • 3.Russell's reduction therefore overcorrects, eliminating sets as a category when restricting comprehension axioms suffices—as Zermelo demonstrated in 1908.
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    Reason against 2 of 2
    ?
    • 1.Russell's type hierarchy requires the axiom of reducibility, which Ramsey and Wittgenstein argued is an empirical assumption smuggled into logic.
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    • 2.If the axiom of reducibility is not a logical truth, the reduction of sets to propositional functions fails to resolve paradoxes on purely logical grounds.
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    Philosophy of Language2 linked

    Related

    Assertions about sets can be reduced to assertions about propositional functions...If the axiom of reducibility is not a logical truth, the reduction of sets to pr...Russell's reduction therefore overcorrects, eliminating sets as a category when ...Russell's type hierarchy requires the axiom of reducibility, which Ramsey and Wi...
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    The restriction that a function of one type cannot apply to a function of the sa...When an alternative framework resolves paradoxes with weaker ontological revisio...Zermelo's axiomatic set theory resolves the same paradoxes without eliminating s...

    Similar

    Assertions about sets can be reduced to assertions about propositional...90%The discovery of paradoxes in early set theories gave mathematicians a...82%There have been attempts to avoid Russell's paradox by altering the un...81%The inconsistency of set theory threatens the trustworthiness of all m...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: principia-mathematica
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    The paradoxes of the theory of sets are resolved by reducing assertions about sets to assertions about propositional functions. The restriction that a function of one type cannot apply to a function of the same type is enough to block the paradoxes. Thus the distinction between individuals, functions of individuals, and functions of such functions, categorized by what came to be called “simple theory of types” is enough for the purposes of reducing mathematics to classes, and so to logic. The id
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    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit