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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Admitting Omega as an ordinal within On generates Burali-Forti paradox: Omega would be less than itself, since On is well-ordered by membership.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Modern set theory (ZFC) avoids this by distinguishing: On is a proper class, not a set, so Omega ∉ On by design.
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    • 2.No paradox occurs when Omega is correctly understood as the supremum of all ordinals rather than a member among them.
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    • 3.The Burali-Forti issue reflects confusion about type distinctions, not a flaw in well-ordering principles themselves.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.If Omega is an ordinal in On, then Omega has an ordinal successor Omega+1 that is also in On by closure.
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    • 2.Since On is well-ordered by membership, Omega < Omega+1, making Omega a proper initial segment of itself.
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    • 3.This circularity—where Omega generates elements larger than itself within its own domain—constitutes a genuine paradox.
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