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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that Two infinite well-ordered sets with the same ordinal number have the same cardinal number.

    ?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.Ordinal identity guarantees structural isomorphism but not cardinality equivalence when cardinal and ordinal number theories are defined independently.
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      Think about whether this reason is strong or weak

    • 2.In certain constructivist and predicativist frameworks, the inference from order-isomorphism to bijection requires impredicative comprehension axioms that cannot be assumed without circularity.
      ?

      Think about whether this reason is strong or weak

    • 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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      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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      Think about whether this reason is strong or weak

    • 2.If a bijection exists between two sets, those sets have the same cardinal number.
      ?

      Think about whether this reason is strong or weak

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