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    Carmelics

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    Inverse View

    It is not the case that The cardinality count of second-order sentences presupposes a fixed background set theory, but set-theoretic pluralism (Hamkins) allows different universes where 'countably many' is not absolute.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Cardinality comparisons (countable vs. uncountable) remain invariant across set-theoretic universes sharing the same ordinal structure.
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    • 2.Second-order logic can be given semantics independent of any single background universe (category-theoretic or proof-theoretic approaches).
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    • 3.Pluralism about set theory doesn't entail pluralism about logical consequence; countability is definable uniformly across models.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Set-theoretic pluralism (Hamkins) demonstrates that cardinality is relative to model; what counts as 'countable' varies across universes.
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    • 2.Second-order logic's semantics require fixing a background universe to determine which sets satisfy quantified formulas.
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    • 3.If pluralism is true, claims about 'all second-order sentences' presuppose a particular universe rather than an absolute standard.
      ?

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