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    Carmelics

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

    It is not the case that Any sufficiently strong formal theory F satisfying the conditions of the first incompleteness theorem must possess non-standard models in addition to its intended standard model.

    ?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 existence of non-standard models is a semantic fact about model theory, but the incompleteness theorems are syntactic results about provability within formal systems.
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      Think about whether this reason is strong or weak

    • 2.Conflating syntactic unprovability with semantic model-theoretic multiplicity commits a use-mention error: G_F being unprovable in F does not entail F lacks a unique intended interpretation, only that F cannot prove it.
      ?

      Think about whether this reason is strong or weak

    • 3.Hilbert's distinction between formal systems and their intended domains supports the view that the standard model N is fixed by our pre-formal grasp of the natural numbers, not by any axiom set.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Kreisel's categoricity argument holds that second-order Peano Arithmetic is categorical, uniquely determining the standard model up to isomorphism despite incompleteness results applying to its first-order fragments.
      ?

      Think about whether this reason is strong or weak

    • 2.If a second-order formulation of F can fix the standard model categorically, then the existence of non-standard models of first-order F reflects an expressive limitation of first-order logic, not an ineliminable feature of F's intended semantics.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.If a theory F has independent statements (such as the Gödel sentence G_F), then F must have models satisfying G_F and models satisfying ¬G_F.
      ?

      Think about whether this reason is strong or weak

    • 2.A theory cannot simultaneously rule out all non-intended interpretations and fix a unique intended interpretation if independent statements exist.
      ?

      Think about whether this reason is strong or weak

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