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    There are structures of every infinite cardinality which ... — Carmelics
    Home/Skepticism
    HistoryEditSee Inverse

    There are structures of every infinite cardinality which are not second-order characterizable.

    Modality & Possibility
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    2 reasons against

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    Reasons For

    1 perspective
    Reason for
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    • 1.There are only countably many second-order sentences.
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    • 2.Therefore there are only countably many (up to isomorphism) second-order characterizable structures.
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    • 3.There exist structures of every infinite cardinality.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The cardinality count of second-order sentences presupposes a fixed background set theory, but set-theoretic pluralism (Hamkins) allows different universes where 'countably many' is not absolute.
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    • 2.If second-order logic is interpreted with full semantics relative to a given set-theoretic universe, what counts as 'all structures' shifts across universes, undermining the fixed comparison between sentence-count and structure-count.
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    • 3.Therefore the argument's P2-P4 inference is valid only relative to a single set-theoretic universe, not absolutely, weakening the modal force of the original claim.
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    Reason against 2 of 2
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    • 1.Henkin semantics for second-order logic permits non-standard interpretations of quantifiers over relations, under which second-order logic is complete and its Löwenheim-Skolem theorems hold (Henkin 1950).
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    • 2.Under Henkin semantics, the argument that only countably many structures are characterizable fails, because the expressive power of second-order logic collapses to that of many-sorted first-order logic, which characterizes structures up to elementary equivalence rather than isomorphism.
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    • 3.The original claim tacitly assumes full semantics, but the choice of semantics is a substantive philosophical commitment, not a logical given, so the claim is semantics-relative rather than an absolute logical fact.
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    Topics

    SkepticismModality & Possibility

    Connections

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    Truth & Knowledge3 linked

    Related

    Countably many structures cannot exhaust all structures of every infinite cardin...Henkin semantics for second-order logic permits non-standard interpretations of ...If second-order logic is interpreted with full semantics relative to a given set...The cardinality count of second-order sentences presupposes a fixed background s...
    +6 moreShow less
    The original claim tacitly assumes full semantics, but the choice of semantics i...There are only countably many second-order sentences.There exist structures of every infinite cardinality.Therefore the argument's P2-P4 inference is valid only relative to a single set-...Therefore there are only countably many (up to isomorphism) second-order charact...Under Henkin semantics, the argument that only countably many structures are cha...

    Similar

    There exist structures of every infinite cardinality.83%Therefore there are only countably many (up to isomorphism) second-ord...83%Countably many structures cannot exhaust all structures of every infin...82%Infinite structures cannot be fully characterized by deductive means i...79%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    Are all structures second-order characterizable? There are only countably many second-order sentences, hence only countably many (up to isomorphism) second-order characterizable structures. Therefore there are lots of structures of every infinite cardinality which are not second-order characterizable. However, it is not easy to give examples. One example is \((\kappa,<)\), where \(\kappa\) is the first measurable cardinal (\(>\omega\)). See §7.2 for an explanation. Another example is \((
    Extraction notes

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

    Details

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