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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
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    Home/Original/inverse
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    Inverse View

    It is not the case that Formal deductive systems like Peano arithmetic generate theorems that their axioms do not explicitly contain, as Gödel's incompleteness results demonstrate.

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

    Reasons For

    1 perspective
    Reason for
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    • 1.Unprovable statements like the Gödel sentence are arguably *contained* in axioms as logical consequences, just not derivable by finite proof.
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    • 2.The incompleteness theorems address what's *provable*, not what's *contained*; containment is a semantic, not syntactic, property.
      ?

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    • 3.Even unprovable truths follow necessarily from axioms, so claiming they're 'not contained' conflates logical derivability with conceptual generation.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Gödel proved PA cannot prove all true statements about natural numbers, demonstrating genuine limits to axiomatic systems.
      ?

      Think about whether this reason is strong or weak

    • 2.Theorems derived through valid inference rules represent knowledge not present in axioms alone, only implicit in their logical consequences.
      ?

      Think about whether this reason is strong or weak

    • 3.The incompleteness theorems reveal that formal systems cannot be simultaneously complete, consistent, and effectively axiomatizable.
      ?

      Think about whether this reason is strong or weak

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