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    Home/Original/inverse
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    Inverse View

    It is not the case that Truth in second-order logic ('M ⊨_s φ') is not an absolute property relative to ZFC.

    ?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 non-absoluteness of second-order truth relative to ZFC reflects a limitation of ZFC as a metatheory, not an intrinsic property of second-order truth itself.
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    • 2.Kreisel's squeezing argument and Shapiro's categoricity results suggest second-order logical consequence has a determinate extension fixed by the standard semantics, independent of set-theoretic indeterminacy.
      ?

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    • 3.If we adopt a background ontology of full powersets as primitive (as in Zermelo's 1930 quasi-categoricity results), CH has a determinate truth value, making second-order truth absolute relative to that stronger framework.
      ?

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    Reason for 2 of 2
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    • 1.The claim conflates syntactic provability within ZFC with semantic absoluteness, since a sentence can be semantically determinate even when ZFC cannot decide it.
      ?

      Think about whether this reason is strong or weak

    • 2.Gödel's platonist position, endorsed in 'What is Cantor's Continuum Problem?', holds that set-theoretic statements like CH possess objective truth values that human axiom systems may simply fail to capture.
      ?

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    • 3.The ZFC-relative non-absoluteness of 'M ⊨_s φ' therefore demonstrates only epistemic incompleteness, not genuine ontological indeterminacy in second-order truth.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.For first-order logic, the satisfaction relation 'M ⊨_s φ' is absolute relative to ZFC and can be written in Δ₁^ZFC form.
      ?

      Think about whether this reason is strong or weak

    • 2.For second-order logic, while 'φ is a second-order formula', 'M is an L-structure', and 's is an assignment' are all absolute relative to ZFC, the satisfaction relation 'M ⊨_s φ' is not absolute.
      ?

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    • 3.The truth of the second-order sentence θ_CH in a sufficiently large model depends on whether the Continuum Hypothesis holds, and the Continuum Hypothesis is not settled by ZFC alone.
      ?

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