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    A proof of natural number closure under Precedes that rel... — Carmelics
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    Challenges→If Precedes(n,a) and n is a natural number, then a is a natural number

    A proof of natural number closure under Precedes that relies on impredicative second-order comprehension cannot establish the claim on purely logical grounds alone.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Impredicative comprehension quantifies over all second-order objects, including those defined via the very comprehension principle being invoked.
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    • 2.Logical grounds alone concern what follows from logical axioms and rules without mathematical assumptions about infinite totalities.
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    • 3.Circularity in defining a set via quantification over a domain that includes the set itself exceeds what pure logic permits independently.
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    Reasons Against

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    Reason against
    ?
    • 1.Second-order logic with impredicative comprehension is a formal system with fully specified axioms, making proofs within it logically rigorous.
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    • 2.The distinction between 'logical' and 'mathematical' grounds is itself contested; second-order logic enjoys mainstream logical status.
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    • 3.Predicativity restrictions are methodological preferences, not logical requirements, so impredicative proofs remain logically valid.
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    Connections

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    Proof of definition segments1 linkedTruth & Knowledge1 linked

    Related

    Circularity in defining a set via quantification over a domain that includes the...If Precedes(n,a) and n is a natural number, then a is a natural numberImpredicative comprehension quantifies over all second-order objects, including ...Logical grounds alone concern what follows from logical axioms and rules without...
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    Predicativity restrictions are methodological preferences, not logical requireme...Second-order logic with impredicative comprehension is a formal system with full...The distinction between 'logical' and 'mathematical' grounds is itself contested...

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