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    Predicativist analysis is a conservative extension of Pea... — Carmelics
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    Challenges→Quine and Putnam's indispensability arguments commit us at most to what Peano Arithmetic commits us to

    Predicativist analysis is a conservative extension of Peano Arithmetic

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Feferman gave a detailed formal presentation of predicativist analysis
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    • 2.Feferman proved that, on a certain reconstruction, the predicativist theory does not prove any new arithmetical statements beyond those provable in Peano Arithmetic
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Feferman's reconstruction encodes specific philosophical commitments about predicativity that are themselves contested, not neutral formal choices.
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    • 2.Alternative reconstructions of predicativist analysis, such as those explored by Weyl, yield systems with different proof-theoretic strengths than PA.
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    • 3.A conservative extension result is only as robust as the reconstruction it presupposes, so disputed reconstructions yield disputed conservation results.
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    Reason against 2 of 2
    ?
    • 1.Proof-theoretic conservation over PA does not establish that predicativist analysis is semantically or ontologically conservative over arithmetic.
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    • 2.Predicativist analysis quantifies over real numbers and function spaces whose existence is not warranted by the ontology presupposed in Peano Arithmetic.
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    • 3.A theory can be deductively conservative yet introduce new entities and concepts that represent genuine extensions of the original framework's commitments.
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    Related

    A conservative extension result is only as robust as the reconstruction it presu...A theory can be deductively conservative yet introduce new entities and concepts...All mathematics used in physical theory can be recaptured in a predicative syste...Alternative reconstructions of predicativist analysis, such as those explored by...
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    Feferman gave a detailed formal presentation of predicativist analysisFeferman proved that, on a certain reconstruction, the predicativist theory does...Feferman's reconstruction encodes specific philosophical commitments about predi...Predicativist analysis quantifies over real numbers and function spaces whose ex...Proof-theoretic conservation over PA does not establish that predicativist analy...Quine and Putnam's indispensability arguments commit us at most to what Peano Ar...Therefore indispensability arguments from physics do not require ontological com...

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    An intermediate position is that defended by classical “predicativists” such as Poincaré and Weyl. The theory, presented in a satisfactory logical way by Feferman and others, accepts the excluded middle on the natural numbers (and as such it is arguably committed to the existence of the set of natural numbers and in any case to accepting bivalence on the natural numbers) but does not accept the existence of the power set of the natural numbers. According to predicativism (see Feferman 2005), set

    Details

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