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    Two infinite well-ordered sets with the same ordinal numb... — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    Two infinite well-ordered sets with the same ordinal number have the same cardinal number.

    Modality & Possibility
    ?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.Two sets well-ordered with the same order type have a unique correspondence between elements in corresponding positions of the ordering.
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    • 2.If a bijection exists between two sets, those sets have the same cardinal number.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Ordinal identity guarantees structural isomorphism but not cardinality equivalence when cardinal and ordinal number theories are defined independently.
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    • 2.In certain constructivist and predicativist frameworks, the inference from order-isomorphism to bijection requires impredicative comprehension axioms that cannot be assumed without circularity.
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    • 3.Poincaré and Weyl both argued that classical transfinite arithmetic conflates structural correspondence with genuine numerical equality, making the entailment from same ordinal to same cardinal non-trivial.
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    Reason against 2 of 2
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    • 1.The claim holds only within ZFC set theory; in alternative foundations like NFU or Aczel's non-well-founded set theory, the relationship between ordinal and cardinal structure is not preserved in the same way.
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    • 2.Philosophical accounts of cardinality grounded in Fregean abstraction principles, such as Hume's Principle, treat cardinal number as conceptually prior to and independent of ordinal structure, severing the entailment the argument assumes.
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    Topics

    Modality & PossibilityProof of definition segments

    Related

    If a bijection exists between two sets, those sets have the same cardinal number...In certain constructivist and predicativist frameworks, the inference from order...Ordinal identity guarantees structural isomorphism but not cardinality equivalen...Philosophical accounts of cardinality grounded in Fregean abstraction principles...
    +3 moreShow less
    Poincaré and Weyl both argued that classical transfinite arithmetic conflates st...The claim holds only within ZFC set theory; in alternative foundations like NFU ...Two sets well-ordered with the same order type have a unique correspondence betw...

    Similar

    Not all infinite sets have the same cardinality.86%The natural order on ordinal numbers is a well-ordering and a set in N...78%There is no set of all cardinal numbers and no set of all ordinal numb...77%If a bijection exists between two sets, those sets have the same cardi...77%

    Source

    AI-extracted1/3 agreementValid
    SEP: infinity
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    A cardinal number (like “one”, “two”, “three”)—also called a cardinality—represents how many elements a set has. Two sets have the same cardinal number if it is possible to come up with any correspondence at all between the members of one and the members of the other, even if this correspondence fails to respect the ordering or any other structure of the sets. Two finite sets have the same cardinal number if and only if they have the same ordinal number. For infinite sets, if they are well-order
    Extraction notes

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

    Details

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