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    Every cardinal number can be represented by an ordinal nu... — Carmelics
    Home/Modality & Possibility
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    Part of a larger discussion

    Supports→The cardinal numbers must be well-ordered

    Every cardinal number can be represented by an ordinal 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.Cantor's well-ordering principle states that every set can be put into some well-ordered form
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    • 2.The well-ordering principle is equivalent to the Axiom of Choice
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    • 3.The definition of ordinal numbers ensures that for any non-empty set of ordinal numbers there is always a first element
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The Well-Ordering Theorem is not a logical truth but a contested axiom equivalent to AC, which many constructivists reject.
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    • 2.Without AC, there exist sets (e.g., certain infinite subsets of the reals) that cannot be well-ordered, blocking ordinal representation.
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    • 3.If cardinal representation requires AC as a necessary condition, the claim is not a theorem of ZF but a conditional on a disputed axiom.
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    Reason against 2 of 2
    ?
    • 1.Proper classes like the class of all cardinals resist set-theoretic treatment, as Burali-Forti's paradox demonstrates for the totality of ordinals.
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    • 2.If every cardinal required an ordinal representative, the totality of such ordinals would itself constitute an ordinal, generating a contradictory greatest ordinal.
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    Topics

    Modality & Possibility

    Related

    Cantor's well-ordering principle states that every set can be put into some well...For any non-empty set of ordinal numbers there is always a first ordinal numberIf cardinal representation requires AC as a necessary condition, the claim is no...If every cardinal required an ordinal representative, the totality of such ordin...
    +6 moreShow less
    Proper classes like the class of all cardinals resist set-theoretic treatment, a...The Well-Ordering Theorem is not a logical truth but a contested axiom equivalen...The cardinal numbers must be well-orderedThe definition of ordinal numbers ensures that for any non-empty set of ordinal ...The well-ordering principle is equivalent to the Axiom of ChoiceWithout AC, there exist sets (e.g., certain infinite subsets of the reals) that ...

    Similar

    There is no set of all cardinal numbers and no set of all ordinal numb...83%For every ordinal number there is a corresponding aleph cardinal81%The operation T on ordinal numbers can be defined in NFU78%The definition of ordinal numbers ensures that for any non-empty set o...78%

    Source

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    SEP: infinity
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    Thus, just as there is an infinite hierarchy of infinite ordinal numbers, which Cantor represented with lowercase Greek letters, there is also an infinite hierarchy of infinite cardinal numbers, which Cantor represented with Hebrew letters, and in particular aleph, “\(\aleph\)”. The finite cardinals are 0, 1, 2, 3, …. The first infinite cardinal, that of the natural numbers (and all countable sets), is \(\aleph_0\). Cantor’s “well-ordering principle”, stating that every set can be put into some

    Details

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