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    Internal categoricity provides a bridge between full sema... — Carmelics
    Home/Truth & Knowledge
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    Internal categoricity provides a bridge between full semantics and Henkin semantics, showing that full semantics does not monopolize categoricity.

    Proof of definition segmentsTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Internal categoricity is defined proof-theoretically and therefore behaves the same way in both full and Henkin models.
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    • 2.Full semantics is a limit case of Henkin semantics.
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    • 3.Classical second-order sentences characterizing arithmetic and analysis are internally categorical under both interpretations.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Internal categoricity proofs presuppose a metatheory in which 'all models' is itself interpreted, reintroducing the same semantic commitments.
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    • 2.Väänänen and Wang (2015) demonstrate that internal categoricity results are only as strong as the background set-theoretic assumptions sustaining the metatheory.
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    • 3.Therefore, internal categoricity does not escape but merely relocates the dependence on a privileged semantic framework.
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    Reason against 2 of 2
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    • 1.Henkin semantics permits non-standard comprehension axioms, meaning 'categoricity' within Henkin models tracks structural properties relative to those axiom schemes, not an absolute notion.
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    • 2.A result showing categoricity across both full and Henkin semantics conflates two distinct senses of categoricity that differ in their mathematical and philosophical import.
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    • 3.Distinguishing these senses, as Shapiro's 1991 work on foundations of mathematics demands, reveals that the bridge claim equivocates rather than unifies.
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    Topics

    Truth & KnowledgeProof of definition segments

    Connections

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    Philosophy of Language1 linked

    Related

    A result showing categoricity across both full and Henkin semantics conflates tw...Classical second-order sentences characterizing arithmetic and analysis are inte...Distinguishing these senses, as Shapiro's 1991 work on foundations of mathematic...Full semantics is a limit case of Henkin semantics.
    +5 moreShow less
    Henkin semantics permits non-standard comprehension axioms, meaning 'categoricit...Internal categoricity is defined proof-theoretically and therefore behaves the s...Internal categoricity proofs presuppose a metatheory in which 'all models' is it...Therefore, internal categoricity does not escape but merely relocates the depend...Väänänen and Wang (2015) demonstrate that internal categoricity results are only...

    Similar

    Internal categoricity is defined proof-theoretically and therefore beh...88%Internal categoricity is stronger than categoricity.84%Internal categoricity is stronger than (mere) categoricity.84%A categorical theory is semantically complete: for every sentence φ in...84%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    Solovay has made a posting in FOM (2006 Other Internet Resources) in which he shows that the related statement that every complete second-order sentence \(\theta\) is categorical, is independent of ZFC. There is a strong form of categoricity which holds for Henkin structures in important cases and agrees with the usual concept of categoricity in the case of full Henkin models. It builds on the remarkable ability of second-order logic to express its own categoricity. The isomorphism \((M,R)\co
    Extraction notes

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

    Details

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