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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that 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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    Reasons For

    1 perspective
    Reason for
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    • 1.The invariant content of a sentence under full semantics persists across universes: satisfaction relations remain coherent under elementary embeddings.
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    • 2.Set-theoretic pluralism doesn't undermine logic; we can compare universes using a metatheory that relativizes quantifiers without loss of meaning.
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    • 3.Second-order expressiveness is captured by what sentences can distinguish structurally, which is independent of which universe counts as 'all'.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Full second-order semantics quantifies over all subsets relative to a universe V; different set-theoretic universes have different power sets.
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    • 2.Completeness theorems depend on fixing a background universe; changing universes changes which structures satisfy a sentence.
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    • 3.Without absolute universe-independence, 'all structures' is not an absolute notion, making cross-universe comparison of logical strength unstable.
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