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    The cardinal numbers must be well-ordered — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    The cardinal numbers must be well-ordered

    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.Every cardinal number can be represented by an ordinal number
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    • 2.For any non-empty set of ordinal numbers there is always a first ordinal number
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The claim that every cardinal can be represented by an ordinal presupposes the Well-Ordering Theorem, which is independent of ZF set theory.
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    • 2.Accepting the Axiom of Choice—required to derive the Well-Ordering Theorem—is a substantive mathematical commitment, not a logical necessity.
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    • 3.Without AC, there exist models of set theory containing cardinals that cannot be well-ordered, making the original claim contingent rather than necessary.
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    Reason against 2 of 2
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    • 1.Constructivist and intuitionist mathematicians like Brouwer reject the classical conception of completed infinite totalities on which ordinal representation depends.
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    • 2.If infinite cardinals are not completed totalities but potential infinities, the second premise—that any non-empty set of ordinals has a least element—cannot be straightforwardly applied.
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    Modality & Possibility

    Related

    Accepting the Axiom of Choice—required to derive the Well-Ordering Theorem—is a ...Constructivist and intuitionist mathematicians like Brouwer reject the classical...Every cardinal number can be represented by an ordinal numberFor any non-empty set of ordinal numbers there is always a first ordinal number
    +3 moreShow less
    If infinite cardinals are not completed totalities but potential infinities, the...The claim that every cardinal can be represented by an ordinal presupposes the W...Without AC, there exist models of set theory containing cardinals that cannot be...

    Similar

    Ordinal numbers designate each cardinal's position in their well-order...88%Two infinite well-ordered sets with the same ordinal number have the s...76%Every cardinal number can be represented by an ordinal number75%The natural order on ordinal numbers is a well-ordering and a set in N...75%

    Source

    AI-extracted1/3 agreementValid
    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
    Extraction notes

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

    Details

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