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    The formal accommodation of proper classes solves no math... — Carmelics
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    Challenges→Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set theory formally accommodate proper classes, including the class of all cardinals, as legitimate mathematical objects with determinate extensions.

    The formal accommodation of proper classes solves no mathematical problems ZFC cannot solve through alternative methods, making it ontologically extravagant without epistemic gain.

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    Key Terms

    Epistemic gain(what the statement says isn't achieved)
    An actual improvement in what we can know or how reliably we can know it; genuine progress in understanding.
    Formal accommodation(as used in mathematical philosophy)
    A way of officially recognizing or including something within a system of rules or theory.
    Ontologically extravagant(as used in metaphysics)
    Requiring too many or unnecessarily complicated things to actually exist in reality; 'ontologically' means concerning what exists, and 'extravagant' means excessive or wasteful.
    Ontology/Ontological(in metaphysics)
    The philosophical study of what actually exists or is real, as opposed to what merely seems to exist or what we can know about things.
    Proper classes

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    (as used in set theory)
    In set theory, collections of things that are too large or too weird to be treated as ordinary 'sets' (think of them as super-big collections).
    ZFC(Classical set theory as a foundation for mathematics)
    The axiom system ZF plus the axiom of choice (AC).
    epistemology/epistemic(the 'epistemic' in 'epistemically determinate')
    Epistemology is the study of knowledge and how we know things. 'Epistemic' means 'related to knowledge or knowing.'

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    Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set the...

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    Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set the...

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