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    A semantic argument in set theory is convertible to a syn... — Carmelics
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    Home/Philosophy of Language
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    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

    Philosophy of LanguageTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.The Completeness Theorem is applicable only when a semantic argument works in whatever model of set theory is being used
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    • 2.Model-specific properties such as CH, ◇, or 2^ℵ₀ = ℵ₂ must be stated separately as assumptions when used
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    • 3.Non-first-order properties of models cannot be used at all if the Completeness Theorem is to apply
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.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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    • 2.The claim smuggles in a privileged notion of 'all models of ZFC' that cannot be cashed out without a metatheory subject to the same underdetermination.
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    • 3.Hilbert and Bernays demonstrated that syntactic derivability and semantic validity come apart precisely when the metatheory is not fixed independently of the object theory.
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    Reason against 2 of 2
    ?
    • 1.Forcing constructions, as developed by Cohen, produce extensions of ground models where new syntactic proofs become available that were unavailable in the original model.
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    • 2.A semantic argument valid only in a forcing extension is nonetheless convertible to a syntactic proof relative to the axioms of that extension, which contradicts the claim's universal quantification over all ZFC models.
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    Philosophy of LanguageTruth & Knowledge

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    Modality & Possibility1 linkedSkepticism1 linked

    Related

    A semantic argument valid only in a forcing extension is nonetheless convertible...Forcing constructions, as developed by Cohen, produce extensions of ground model...Gödel's completeness theorem applies to first-order logic, but ZFC's intended se...Hilbert and Bernays demonstrated that syntactic derivability and semantic validi...
    +4 moreShow less
    Model-specific properties such as CH, ◇, or 2^ℵ₀ = ℵ₂ must be stated separately ...Non-first-order properties of models cannot be used at all if the Completeness T...The Completeness Theorem is applicable only when a semantic argument works in wh...

    Similar

    The Completeness Theorem is applicable only when a semantic argument w...87%The semantic argument must be valid in all Henkin models, not only in ...84%Second-order logic satisfies the Completeness Theorem when Henkin mode...82%A semantic argument is valid if and only if every interpretation that ...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
    View source passageHide passage
    Let \(c,d\in (a,b)\) such that \(f(c)<0\) and \(f(d)>0\). Without loss of generality, \(c<d\). Let \(X=\{e\in(a,b) : f(e)<0\}\). Since we have relation variables for subsets of the domain, we can think of X simply as a value of such a relation variable. , \(X=\{e : e\notin X\}\)) and then we should not be able to claim that it exists. However, in this case the Comprehension Axiom Schema implies that X exists. Clearly, \(X\ne\emptyset\) and X is bounded from above by d. One of the sec
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    The claim smuggles in a privileged notion of 'all models of ZFC' that cannot be ...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit