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    Carmelics

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

    It is not the case that Non-standard models of F must contain 'infinite' non-natural numbers beyond all natural numbers.

    ?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 term 'infinite non-natural numbers' imports set-theoretic ontology into a purely syntactic result about provability within formal systems.
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    • 2.Non-standard models are model-theoretic artifacts relative to a metatheory; their 'non-standard elements' need not be reified as numbers of any kind.
      ?

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    • 3.Skolem's original (1933) construction of non-standard arithmetic was intended to show the indeterminacy of the natural number concept, not to populate models with a new category of object.
      ?

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    Reason for 2 of 2
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    • 1.The argument presupposes that 'strongly represents' the proof relation in F suffices to transfer facts about standard numerals to all elements of non-standard models, but this inference requires the metatheory to already privilege the standard model.
      ?

      Think about whether this reason is strong or weak

    • 2.On a strict formalist reading (cf. Curry, Detlefsen's 'Hilbert's Program'), there is no well-defined notion of 'natural number' outside a formal system against which model elements can be measured as 'non-standard'.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.¬G_F is equivalent to ∃x Prf_F(x, ⌈G_F⌉), so models satisfying ¬G_F must contain entities witnessing the formula Prf_F(x, ⌈G_F⌉).
      ?

      Think about whether this reason is strong or weak

    • 2.Because Prf_F(x, y) strongly represents the proof relation, F can prove ¬Prf_F(n̲, ⌈G_F⌉) for every standard numeral n̲.
      ?

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    • 3.Therefore no natural number n can witness the formula Prf_F(x, ⌈G_F⌉) in any non-standard model.
      ?

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