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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
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    42
    Home/Original/inverse
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    Inverse View

    It is not the case that There are structures of every infinite cardinality which are not second-order characterizable.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 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).
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.There are only countably many second-order sentences.
      ?

      Think about whether this reason is strong or weak

    • 2.Therefore there are only countably many (up to isomorphism) second-order characterizable structures.
      ?

      Think about whether this reason is strong or weak

    • 3.There exist structures of every infinite cardinality.
      ?

      Think about whether this reason is strong or weak

    Next step

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    Strongest counterpoint
    Explore the most compelling reason on the other side.