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    For any two well-ordered sets, all positions of one well-... — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    For any two well-ordered sets, all positions of one well-ordering must correspond to initial positions in the other well-ordering.

    Modality & PossibilityProof of definition segments
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Topics

    Modality & PossibilityProof of definition segments

    Connections

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    Truth & Knowledge1 linked

    Related

    But if the first non-corresponding elements correspond, this contradicts the ass...But the claim that the subset of non-corresponding positions is itself well-orde...Cantor's original proof requires transfinite induction, whose legitimacy over al...Constructivists like Brouwer deny that a non-empty set of positions must contain...
    +5 moreShow less
    For any two well-ordered sets, initial positions in one ordering correspond to i...If not all positions of one well-ordering corresponded to initial positions in t...If the set of non-corresponding positions were non-empty, the first elements of ...The argument assumes any non-empty subset of non-corresponding positions has a f...The reductio in P3-P4 relies on classical excluded middle applied to infinite we...

    Similar

    For any two well-ordered sets, initial positions in one ordering corre...94%All positions of one well-ordering must correspond to initial position...94%If not all positions of one well-ordering corresponded to initial posi...90%The correspondence of positions across all well-ordered sets is total ...82%

    Source

    AI-extracted1/3 agreementValid
    SEP: infinity
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    Cantor noted that for any two well-ordered sets, the initial positions in one ordering (the first, the second, the third, etc.) correspond to the initial positions in the other, the way that they do for finite sets. In fact, he showed that all of the positions of one well-ordering must correspond to initial positions in the other. (If this weren’t true, then the set of positions in one that don’t correspond to positions in the other would be non-empty for each set, and the first elements of thes
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit