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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that A formula can be proof-theoretically valid without being derivable in any given formal system, as shown by incompleteness phenomena affecting extensions of PA.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
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    • 1.The distinction between 'proof-theoretically valid' and 'derivable' conflates semantic truth with proof-theoretic properties confusingly.
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    • 2.Any coherent notion of 'proof-theoretic validity' must ultimately be definable within or relative to some formal system's rules.
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    • 3.Incompleteness shows limits of specific systems, not that validity exists outside all possible formal derivation frameworks.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Gödel's incompleteness theorems prove true sentences exist unprovable in any consistent formal system extending PA.
      ?

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    • 2.Proof-theoretic validity concerns what follows from axioms; derivability concerns what a specific system can derive—these diverge.
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    • 3.We can semantically recognize formulas as valid through model-theoretic means even when no proof exists in formal systems.
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