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    A property that can be destroyed or created by set-theore... — Carmelics
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    Challenges→A weakly compact inaccessible cardinal is exceedingly large

    A property that can be destroyed or created by set-theoretic forcing does not robustly ground claims about absolute cardinal magnitude.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Forcing can change whether certain sets have specific cardinalities (e.g., Cohen reals alter continuum size), yet the underlying set-theoretic universe remains consistent.
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    • 2.If cardinal magnitude were absolute, no forcing extension could modify cardinality facts without contradiction. Forcing's consistency shows cardinality is model-dependent.
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    • 3.Properties robust to all ground-theoretic variation would constrain mathematical possibility in ways forcing demonstrates they do not.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Forcing preserves cardinality relations within each model; what changes is which model we inhabit, not whether facts hold absolutely in that model.
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    • 2.A property can be robust (determined by basic axioms) even if multiple consistent extensions exist, just as physical laws are robust despite multiple possible worlds.
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    • 3.Forcing's existence-compatibility shows pluralism about models, not that cardinal magnitude lacks determinate grounding within each model's structure.
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    Connections

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    Modality & Possibility1 linked

    Related

    A property can be robust (determined by basic axioms) even if multiple consisten...A weakly compact inaccessible cardinal is exceedingly largeForcing can change whether certain sets have specific cardinalities (e.g., Cohen...Forcing preserves cardinality relations within each model; what changes is which...
    +3 moreShow less
    Forcing's existence-compatibility shows pluralism about models, not that cardina...If cardinal magnitude were absolute, no forcing extension could modify cardinali...Properties robust to all ground-theoretic variation would constrain mathematical...

    Details

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    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit