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    A semantic argument valid only in a forcing extension is ... — 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

    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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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
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    • 1.Forcing extensions are conservative over ZFC: any sentence provable in an extension is already syntactically derivable from ZFC plus extension axioms.
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    • 2.Universal quantification over all ZFC models is meaningless if semantic validity shifts between models; local syntactic proofs respect this model-relativity.
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    • 3.The claim conflates 'valid in all ZFC models' with 'valid in all models of extended theory', revealing an equivocation that syntactic analysis exposes.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Syntactic convertibility doesn't address the core issue: the semantic argument's validity depends on forcing extension structure, not extension axioms alone.
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    • 2.Universal quantification over ZFC models remains intact; the argument simply fails in some models (those not containing the forcing extension), not contradicting universality.
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    • 3.Many semantic insights (like continuum hypothesis independence) resist full reduction to syntactic derivations within any single foundational system.
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    Key Terms

    Universal quantification(as used in logic)
    A logical statement that claims something is true for ALL cases, without exception.
    ZFC(Classical set theory as a foundation for mathematics)
    The axiom system ZF plus the axiom of choice (AC).
    axioms(Stumpf, 1891)
    Propositions that we assume to be true and necessary, originating in the content of judgments.
    forcing extension(as used in mathematical logic)
    A mathematical construction used in set theory that adds new mathematical objects to an existing system while preserving certain properties.
    models(models of global democracy)
    idealized theoretical constructions designed to express the normative qualities of a democratic system as well as its constitutive institutions
    semantic argument(as used in logic and philosophy of language)
    An argument about what words or statements actually mean, rather than about the logical structure of the statements themselves.
    syntactic proof(as used in logic and mathematics)
    A proof that works by following strict logical rules and symbol manipulation, without needing to think about what those symbols actually mean.

    Connections

    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    A semantic argument in set theory is convertible to a syntactic formal proof onl...Forcing extensions are conservative over ZFC: any sentence provable in an extens...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Many semantic insights (like continuum hypothesis independence) resist full redu...
    Syntactic convertibility doesn't address the core issue: the semantic argument's...
    +3 moreShow less
    The claim conflates 'valid in all ZFC models' with 'valid in all models of exten...Universal quantification over ZFC models remains intact; the argument simply fai...Universal quantification over all ZFC models is meaningless if semantic validity...