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    Formal deductive systems like Peano arithmetic generate t... — Carmelics
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    Challenges→Reasoning (syllogism and induction) cannot by itself produce new knowledge

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

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    Reasons For

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    Reason for
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    • 1.Gödel proved PA cannot prove all true statements about natural numbers, demonstrating genuine limits to axiomatic systems.
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    • 2.Theorems derived through valid inference rules represent knowledge not present in axioms alone, only implicit in their logical consequences.
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    • 3.The incompleteness theorems reveal that formal systems cannot be simultaneously complete, consistent, and effectively axiomatizable.
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    Reasons Against

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    Reason against
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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.
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    Related

    Even unprovable truths follow necessarily from axioms, so claiming they're 'not ...Gödel proved PA cannot prove all true statements about natural numbers, demonstr...Reasoning (syllogism and induction) cannot by itself produce new knowledgeThe incompleteness theorems address what's *provable*, not what's *contained*; c...
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    The incompleteness theorems reveal that formal systems cannot be simultaneously ...Theorems derived through valid inference rules represent knowledge not present i...Unprovable statements like the Gödel sentence are arguably *contained* in axioms...

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