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    Carmelics

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    Inverse View

    It is not the case that Small large cardinal axioms are compatible with V=L, unlike measurable cardinals

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.The consistency of ZFC+φ+V=L establishes only proof-theoretic compatibility, not genuine ontological cohabitation in any intended model.
      ?

      Think about whether this reason is strong or weak

    • 2.Scott's 1961 theorem shows measurable cardinals destroy V=L by producing non-constructible sets, revealing that 'compatibility' conceals a deeper modal distinction between what can be consistently asserted and what can actually obtain simultaneously.
      ?

      Think about whether this reason is strong or weak

    • 3.A philosophically robust notion of compatibility requires satisfaction in a single intended model, not merely in separate formal systems that share no common universe.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.The boundary between 'small' and 'large' large cardinals is itself set-theoretically defined, making the claim epistemically circular when used to ground a philosophical distinction about possibility.
      ?

      Think about whether this reason is strong or weak

    • 2.Woodin's work on the inner model program suggests that sufficiently strong small large cardinals may eventually yield incompatibility with V=L, undermining the stability of the claimed dichotomy.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.For each small large cardinal axiom φ, if ZFC+φ is consistent then ZFC+φ+V=L is also consistent
      ?

      Think about whether this reason is strong or weak

    • 2.ZFC + 'There is a measurable cardinal' proves ¬V=L, making measurable cardinals incompatible with V=L
      ?

      Think about whether this reason is strong or weak

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