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    The proof that the principle of Induction can be derived ... — Carmelics
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    The proof that the principle of Induction can be derived without the axiom of reducibility in a modified theory of types is faulty.

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.The proof proposed in Appendix B purports to derive the principle of Induction without the axiom of reducibility.
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    • 2.The proof contains an error that invalidates the derivation.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Myhill (1974) and Hazen (1983) independently demonstrated that a repair to Appendix B's proof is achievable within a ramified type theory with substitutional quantification.
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    • 2.If the proof's core strategy is salvageable under modest technical revision, the fault is in the execution rather than the foundational claim that reducibility is dispensable for induction.
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    • 3.A faulty proof of a true theorem does not establish the falsity of the theorem, only the inadequacy of that particular derivation.
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    Reason against 2 of 2
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    • 1.Chwistek and later Feferman showed that significant portions of classical mathematics can be recovered in predicative systems that eschew the axiom of reducibility.
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    • 2.If induction is recoverable predicatively in these alternative frameworks, Russell's Appendix B aim may be directionally correct even if the specific proof is technically flawed.
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    • 3.The historical assessment of Appendix B as simply 'faulty' risks conflating a local inferential error with a wholesale refutation of the predicativist research program it initiated.
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    Philosophy of LanguageTruth & Knowledge

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    Related

    A faulty proof of a true theorem does not establish the falsity of the theorem, ...Chwistek and later Feferman showed that significant portions of classical mathem...If induction is recoverable predicatively in these alternative frameworks, Russe...If the proof's core strategy is salvageable under modest technical revision, the...
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    Myhill (1974) and Hazen (1983) independently demonstrated that a repair to Appen...The historical assessment of Appendix B as simply 'faulty' risks conflating a lo...The proof contains an error that invalidates the derivation.The proof proposed in Appendix B purports to derive the principle of Induction w...

    Similar

    The proof proposed in Appendix B purports to derive the principle of I...89%Without the axiom of reducibility, the ramified type hierarchy prevent...84%The ramified theory of types is in tension with classical real analysi...81%The axiom of reducibility is required in Principia Mathematica to allo...80%

    Source

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    SEP: principia-mathematica
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    In the appendix B to the second edition of PM, which was written by Russell, there is a technical discussion of the consequences of abandoning the axiom of reducibility. A faulty proof is proposed to show that the principle of Induction can be derived without using the axiom of reducibilty in a modified theory of types (see Linsky 2011). As Russell points out, however, it is not possible to define real numbers using “Dedekindian” classes of rational numbers without assuming the axiom of reducibi
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    Details

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