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    Admitting Omega as an ordinal within On generates Burali-... — Carmelics
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    Challenges→The order type Omega of the natural order on ordinal numbers is itself one of the ordinal numbers

    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

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    Reason for
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    • 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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    Reasons Against

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    Reason against
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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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    Related

    If Omega is an ordinal in On, then Omega has an ordinal successor Omega+1 that i...Modern set theory (ZFC) avoids this by distinguishing: On is a proper class, not...No paradox occurs when Omega is correctly understood as the supremum of all ordi...Since On is well-ordered by membership, Omega < Omega+1, making Omega a proper i...
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    The Burali-Forti issue reflects confusion about type distinctions, not a flaw in...The order type Omega of the natural order on ordinal numbers is itself one of th...This circularity—where Omega generates elements larger than itself within its ow...

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