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    Any sufficiently strong formal theory F satisfying the co... — Carmelics
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    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.

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 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.
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    • 2.A theory cannot simultaneously rule out all non-intended interpretations and fix a unique intended interpretation if independent statements exist.
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    Reasons Against

    2 perspectives
    Reason against 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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    • 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.
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    • 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.
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    Reason against 2 of 2
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    • 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.
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    • 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.
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    Related

    A theory cannot simultaneously rule out all non-intended interpretations and fix...Conflating syntactic unprovability with semantic model-theoretic multiplicity co...Hilbert's distinction between formal systems and their intended domains supports...If a second-order formulation of F can fix the standard model categorically, the...
    +3 moreShow less
    If a theory F has independent statements (such as the Gödel sentence G_F), then ...Kreisel's categoricity argument holds that second-order Peano Arithmetic is cate...The existence of non-standard models is a semantic fact about model theory, but ...

    Similar

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    Source

    AI-extracted1/3 agreementValid
    SEP: goedel-incompleteness
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    It is illuminating to reflect on the first incompleteness theorem also from the model theoretic perspective—though the theorem itself does not in any way require this. Namely, it is possible to conclude that any theory \(F\) satisfying the conditions of the theorem must possess, in addition to the intended interpretation or “standard model” (in the case of arithmetical theories, the structure of natural numbers), non-intended interpretations or “non-standard models”—that no such theory can rule
    Extraction notes

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    Details

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