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    If we adopt a background ontology of full powersets as pr... — Carmelics
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    Challenges→Truth in second-order logic ('M ⊨_s φ') is not an absolute property relative to ZFC.

    If we adopt a background ontology of full powersets as primitive (as in Zermelo's 1930 quasi-categoricity results), CH has a determinate truth value, making second-order truth absolute relative to that stronger framework.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Second-order logic with full powersets achieves quasi-categoricity: it pins down models up to isomorphism, eliminating Löwenheim-Skolem ambiguity.
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    • 2.If an ontology uniquely determines a mathematical structure, truths about that structure become determinate relative to that framework.
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    • 3.CH's truth value is fixed in V (von Neumann hierarchy with full powersets); adopting this as primitive makes CH's status absolute within that domain.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Quasi-categoricity defines models only up to isomorphism, not intrinsic truth; isomorphic models can differ in CH's truth value interpretation.
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    • 2.Taking full powersets as primitive merely shifts the problem: we still lack canonical justification for why *this* powerset conception is the right one.
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    • 3.Determinacy relative to a framework ≠ absolute determinacy; framework-relativity itself concedes that CH's status depends on prior philosophical choices.
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    Key Terms

    Absolute (in philosophical context)(as used in metaphysics)
    Something that is true or real in itself, independent of any particular framework, perspective, or point of view.
    Continuum Hypothesis (CH)(Used as the central example of an independence result motivating the multiverse view)
    A proposition in set theory whose truth value is independent of the standard ZFC axioms; in the multiverse framework, it holds in some universes and fails in others.
    Ernst Zermelo(as a historical figure in mathematics and logic)
    A German mathematician (1871-1953) who developed foundational work in set theory, which is the mathematical study of collections of objects.
    Ontology(Carnap argues this enterprise is based on a mistake)
    The philosophical discipline that tries to answer hard questions about what there really is.
    Powerset(as used in set theory)
    In mathematics, the collection of all possible subsets of a given set. For example, the powerset of {1, 2} includes {}, {1}, {2}, and {1, 2}.
    Quasi-categoricity(as used in mathematical logic)
    A technical property in logic meaning that a set of rules or axioms describes its subject almost uniquely, with only minor variations possible.
    Second-order logic(as used in mathematical logic)
    A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.
    determinate truth value(supervaluation logic)
    A sentence has a determinate truth value when it is uniformly true or uniformly false across all admissible structures; it lacks a determinate truth value when its truth value varies across admissible structures
    primitive(Used by Merricks to characterize the representational nature of propositions)
    A fact or property that has no further explanation; it cannot be derived from or reduced to anything more fundamental.

    Connections

    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    CH's truth value is fixed in V (von Neumann hierarchy with full powersets); adop...Determinacy relative to a framework ≠ absolute determinacy; framework-relativity...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    If an ontology uniquely determines a mathematical structure, truths about that s...
    Quasi-categoricity defines models only up to isomorphism, not intrinsic truth; i...
    +3 moreShow less
    Second-order logic with full powersets achieves quasi-categoricity: it pins down...Taking full powersets as primitive merely shifts the problem: we still lack cano...Truth in second-order logic ('M ⊨_s φ') is not an absolute property relative to ...