There is no ordinal for the order type of the set of all ... — Carmelics

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There is no ordinal for the order type of the set of all ordinals.

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1 reason for

2 reasons against

Reasons For

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Reason for

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1.If there were an ordinal corresponding to the order type of the set of all ordinals, that ordinal would have to contain all ordinals.
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2.An ordinal that contains all ordinals would have to be larger than itself.
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3.No ordinal can be larger than itself — this is a contradiction.
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Reasons Against

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Reason against 1 of 2

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1.Burali-Forti's paradox presupposes classical set theory's extensional ontology, which paraconsistent logicians like Graham Priest reject as non-obligatory.
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2.In dialetheist frameworks, a proposition and its negation can both be true, so 'Ω is larger than itself' need not constitute a reductio.
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3.The inconsistency of the set of all ordinals is thus a feature of a specific logical framework, not a metaphysically necessary truth about ordinals as such.
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