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    Gödel's completeness theorem applies to first-order logic... — Carmelics
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    Challenges→A semantic argument in set theory is convertible to a syntactic formal proof only if the argument is valid in every model of ZFC, not merely in some preferred model

    Gödel's completeness theorem applies to first-order logic, but ZFC's intended semantics presupposes a background set theory that is itself model-relative.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.Gödel's completeness theorem only guarantees semantic consequence in standard models, not across all possible interpretations of set-theoretic axioms.
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    • 2.ZFC's axioms like Replacement and Foundation presuppose a prior grasp of 'set' that transcends first-order formalization itself.
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    • 3.Different models of ZFC exist (V, L, inner models), showing that set-theoretic background assumptions are not uniquely fixed by formal syntax.
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    Reasons Against

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    Reason against
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    • 1.Gödel's completeness theorem applies to ZFC-as-formalized; the metalanguage used to state the theorem already embeds set-theoretic assumptions.
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    • 2.'Model-relative' is itself imprecise—all mathematical reasoning is model-relative in a trivial sense, undermining the specific claim's force.
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    • 3.ZFC's intended semantics need not require a background theory; the axioms directly constrain what counts as a set within the system.
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    Connections

    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    'Model-relative' is itself imprecise—all mathematical reasoning is model-relativ...A semantic argument in set theory is convertible to a syntactic formal proof onl...Different models of ZFC exist (V, L, inner models), showing that set-theoretic b...Gödel's completeness theorem applies to ZFC-as-formalized; the metalanguage used...
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    Gödel's completeness theorem only guarantees semantic consequence in standard mo...ZFC's axioms like Replacement and Foundation presuppose a prior grasp of 'set' t...ZFC's intended semantics need not require a background theory; the axioms direct...

    Details

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    claim
    Perspectives
    2 (1 for, 1 against)
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