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    Mathematical analysis would collapse if the axiom of redu... — Carmelics
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    Mathematical analysis would collapse if the axiom of reducibility is abandoned.

    Philosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Real numbers cannot be defined using Dedekindian classes without the axiom of reducibility.
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    • 2.Mathematical analysis depends on the definition of real numbers via such classes.
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    Reasons Against

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    Reason against 1 of 2
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    • 1.Ramsey showed that the axiom of reducibility could be replaced by treating propositional functions extensionally, collapsing the ramified hierarchy without sacrificing analytic results.
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    • 2.If an alternative logical foundation preserves the definition of real numbers without reducibility, then analysis does not depend on that axiom essentially.
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    • 3.Ramsey's revision, accepted by later logicists including Carnap, constitutes a historically grounded existence proof that analysis survives reducibility's abandonment.
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    Reason against 2 of 2
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    • 1.Weyl and Brouwer demonstrated that a substantial portion of classical analysis can be reconstructed using predicativist or intuitionistic methods without impredicative axioms.
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    • 2.If core results of analysis survive without reducibility, the claim of total collapse is demonstrably overstated.
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    Related

    If an alternative logical foundation preserves the definition of real numbers wi...If core results of analysis survive without reducibility, the claim of total col...Mathematical analysis depends on the definition of real numbers via such classes...Ramsey showed that the axiom of reducibility could be replaced by treating propo...
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    Ramsey's revision, accepted by later logicists including Carnap, constitutes a h...Real numbers cannot be defined using Dedekindian classes without the axiom of re...Weyl and Brouwer demonstrated that a substantial portion of classical analysis c...

    Similar

    The axiom of reducibility is required in Principia Mathematica to allo...80%The proof that the principle of Induction can be derived without the a...79%A necessary condition claim does not constitute a reductive analysis.79%Concepts can be legitimately understood without reductive analysis.78%

    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

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