Non-standard models of F must contain 'infinite' non-natu... — Carmelics

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Non-standard models of F must contain 'infinite' non-natural numbers beyond all natural numbers.

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Reasons For

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Reason for

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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⌉).
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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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Reasons Against

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Reason against 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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