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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 The cardinal numbers must be well-ordered

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The claim that every cardinal can be represented by an ordinal presupposes the Well-Ordering Theorem, which is independent of ZF set theory.
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    • 2.Accepting the Axiom of Choice—required to derive the Well-Ordering Theorem—is a substantive mathematical commitment, not a logical necessity.
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    • 3.Without AC, there exist models of set theory containing cardinals that cannot be well-ordered, making the original claim contingent rather than necessary.
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    Reason for 2 of 2
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    • 1.Constructivist and intuitionist mathematicians like Brouwer reject the classical conception of completed infinite totalities on which ordinal representation depends.
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    • 2.If infinite cardinals are not completed totalities but potential infinities, the second premise—that any non-empty set of ordinals has a least element—cannot be straightforwardly applied.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Every cardinal number can be represented by an ordinal number
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    • 2.For any non-empty set of ordinal numbers there is always a first ordinal number
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