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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that For any two well-ordered sets, all positions of one well-ordering must correspond to initial positions in the other well-ordering.

    ?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 argument assumes any non-empty subset of non-corresponding positions has a first element, which presupposes well-ordering of that subset.
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    • 2.But the claim that the subset of non-corresponding positions is itself well-ordered is precisely what requires proof, not assumption.
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    • 3.Cantor's original proof requires transfinite induction, whose legitimacy over all ordinals cannot be established without prior acceptance of the well-ordering theorem.
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    Reason for 2 of 2
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    • 1.Constructivists like Brouwer deny that a non-empty set of positions must contain a determinately first element absent an explicit construction.
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    • 2.The reductio in P3-P4 relies on classical excluded middle applied to infinite well-orderings, which intuitionistic logic rejects as a valid proof schema.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.For any two well-ordered sets, initial positions in one ordering correspond to initial positions in the other, as they do for finite sets.
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    • 2.If not all positions of one well-ordering corresponded to initial positions in the other, then the set of non-corresponding positions would be non-empty for each well-ordered set.
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    • 3.If the set of non-corresponding positions were non-empty, the first elements of those sets would themselves correspond to each other.
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