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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Proper classes like the class of all cardinals resist set-theoretic treatment, as Burali-Forti's paradox demonstrates for the totality of ordinals.

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    Reasons For

    1 perspective
    Reason for
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    • 1.NBG set theory successfully formalizes proper classes as primitive objects with well-defined membership and equality relations.
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    • 2.The Burali-Forti paradox reflects naive comprehension assumptions, not proof that ordinal totality is inherently intractable mathematically.
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    • 3.Alternative frameworks (category theory, type theory) treat large collections non-paradoxically without restricting them to non-sets fundamentally.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Burali-Forti's paradox arises when treating the ordinal class as a set: it would be an ordinal larger than itself, which is contradictory.
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    • 2.Cumulative hierarchy (V) only generates sets; proper classes exceed this by definition, so they cannot be elements of further structures.
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    • 3.Allowing proper classes as sets collapses distinctions essential to avoiding paradox—size-based stratification becomes impossible.
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