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    If 'all natural numbers' is not a determinate domain, the... — Carmelics
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    Challenges→For all natural numbers n, F proves the negation of Prf_F(n, ⌜G_F⌝)

    If 'all natural numbers' is not a determinate domain, the universal claim lacks a truth-condition independently of a background theory that itself requires justification.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.The natural numbers lack intrinsic boundaries; what counts as 'all' depends on constructive or set-theoretic assumptions not self-evident.
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    • 2.Universal claims require determinate domains to have truth-values; indeterminate domains make truth-conditions relative to framework choice.
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    • 3.Background theories justifying 'natural numbers' (ZFC, intuition, etc.) are themselves philosophically contested and need independent grounding.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Mathematical domains need not be 'determinate' in a metaphysical sense; formal definitions suffice for truth-conditions within formal systems.
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    • 2.Justification regress is unavoidable everywhere; demanding independent justification of background theory sets an impossible standard for all knowledge.
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    • 3.Universal claims about natural numbers remain truth-apt relative to standard interpretations; indeterminacy at foundations doesn't undermine local truth-conditions.
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    Related

    Background theories justifying 'natural numbers' (ZFC, intuition, etc.) are them...For all natural numbers n, F proves the negation of Prf_F(n, ⌜G_F⌝)Justification regress is unavoidable everywhere; demanding independent justifica...Mathematical domains need not be 'determinate' in a metaphysical sense; formal d...
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    The natural numbers lack intrinsic boundaries; what counts as 'all' depends on c...Universal claims about natural numbers remain truth-apt relative to standard int...Universal claims require determinate domains to have truth-values; indeterminate...

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