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    The boundary between 'small' and 'large' large cardinals ... — Carmelics
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    Challenges→Small large cardinal axioms are compatible with V=L, unlike measurable cardinals

    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.

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    Key Terms

    Epistemically circular(describing a logical problem with the argument)
    A situation where you're trying to prove something, but your proof assumes the very thing you're trying to prove—like trying to prove a book is true by quoting the book itself.
    Possibility (in philosophy)(metaphysics)
    Something that could potentially exist or be true, even if it doesn't actually exist right now.
    Set-theoretically defined(formal mathematics)
    Defined using the rules and language of set theory, which is the mathematical study of collections of objects.
    cardinal numbers(Contrasted with ordinal numbers in the theory of transfinites.)
    Numbers that depend only on how many things there are.
    epistemology(Contrasted with purely descriptive scientific inquiry)

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    A normative enterprise that tells us how we ought to reason from evidence and how we ought to justify our beliefs, as distinct from merely describing how we do reason or justify beliefs
    grounding(Drawn from contemporary metaphysics; proposed as potentially applicable to understanding the foundations of legality.)
    A metaphysical relation in which some entities or facts are more foundational than others, providing a hierarchical structure of the world.
    large cardinals(Contemporary set theory proposes large cardinal axioms as a remedy for the incompleteness of ZFC.)
    Set-theoretic principles positing the existence of very large infinite cardinal numbers, used as candidate new axioms to extend ZFC and potentially settle open questions such as the Continuum Hypothesis.

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    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

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    Small large cardinal axioms are compatible with V=L, unlike measurable cardinals

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