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.

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Actually obtain simultaneously(as used in metaphysics): Something that can genuinely exist or happen at the same time in reality, not just in theory.
Consistently asserted(as used in logic): Something you can claim or state without creating a logical contradiction; it doesn't break the rules of logic.
Measurable cardinals(as used in set theory): A highly abstract type of infinite number in mathematics that is so large and structured it has special properties; think of it as an 'super-infinite' that's bigger and more organized than regular infinities.
Non-constructible sets(as used in set theory): Mathematical objects that cannot be built up gradually from simpler pieces using standard rules; they exist but can't be created through any step-by-step process.
Scott's 1961 theorem(as used in mathematical logic): A mathematical proof discovered by logician Dana Scott showing that a certain type of infinite number (called a measurable cardinal) cannot exist within a specific mathematical framework.
V=L(Set theory; Gödel's constructible universe): The statement asserting that all sets are constructible
modal distinction(Meyronnes's preferred account of the distinction between a haecceity and its associated nature.): A distinction that obtains between a thing and an intrinsic mode of that thing, where the intrinsic mode does not vary the thing's formal definition and does not of itself imply any quiddity or formal definition.

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