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    Both Peano's axioms and Dedekind's axioms become demonstr... — Carmelics
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    Both Peano's axioms and Dedekind's axioms become demonstrable

    Truth & 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
    ?
    • The logical theory of cardinals provides a foundation from which both Peano's axioms and Dedekind's axioms can be derived
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Russell's logical theory of cardinals relies on the axiom of reducibility, which Wittgenstein and others argued is not a logical truth but an empirical assumption.
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    • 2.A derivation built on non-logical axioms does not constitute a logical demonstration in the sense required for logicism's foundational ambitions.
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    Reason against 2 of 2
    ?
    • 1.Benacerraf's identification problem shows that multiple incompatible set-theoretic reductions equally satisfy Peano's axioms, undermining any claim that one derivation is canonical.
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    • 2.If the axioms are demonstrable only relative to an arbitrarily chosen construction, they are not demonstrated in an absolute logical sense but merely shown to be consistent within one model.
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    Related

    A derivation built on non-logical axioms does not constitute a logical demonstra...Benacerraf's identification problem shows that multiple incompatible set-theoret...If the axioms are demonstrable only relative to an arbitrarily chosen constructi...Russell's logical theory of cardinals relies on the axiom of reducibility, which...
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    The logical theory of cardinals provides a foundation from which both Peano's ax...

    Similar

    Peano's axioms follow from those definitions under Frege's approach83%Hume's Principle is powerful enough for the derivation of the Dedekind...83%Scientific theorems are derived from axioms81%If the axioms are true, it follows that propositions exist and have th...79%

    Source

    AI-extracted1/3 agreementValid
    SEP: logical-construction
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    But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano’s and Dedekind’s axioms become demonstrable. (Russell 1903, §241)
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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