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    There is an infinite hierarchy of infinite cardinal numbers — Carmelics
    Home/Modality & Possibility
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    There is an infinite hierarchy of infinite cardinal numbers

    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.There is an infinite hierarchy of infinite ordinal numbers
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    • 2.Ordinal numbers designate each cardinal's position in their well-ordering
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    • 3.For every ordinal number there is a corresponding aleph cardinal
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Cantor's theorem presupposes the existence of power sets, but the axiom of power set is not a logical truth but a set-theoretic stipulation.
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    • 2.Without independent justification for the power set axiom, the hierarchy of cardinals reflects a chosen formalism, not an objective mathematical reality.
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    Reason against 2 of 2
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    • 1.Skolem's paradox demonstrates that any first-order theory of sets has a countable model, making 'uncountable infinity' a relative, model-dependent notion.
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    • 2.If uncountable infinities are only definable relative to a model, the claim that there exists an objective infinite hierarchy of cardinals is undermined by semantic indeterminacy.
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    Modality & Possibility

    Related

    Cantor's theorem presupposes the existence of power sets, but the axiom of power...For every ordinal number there is a corresponding aleph cardinalIf uncountable infinities are only definable relative to a model, the claim that...Ordinal numbers designate each cardinal's position in their well-ordering
    +3 moreShow less
    Skolem's paradox demonstrates that any first-order theory of sets has a countabl...There is an infinite hierarchy of infinite ordinal numbersWithout independent justification for the power set axiom, the hierarchy of card...

    Similar

    There is an infinite hierarchy of infinite ordinal numbers91%There exist structures of every infinite cardinality.79%Each form in the relevant infinite hierarchy is both one and not one77%There are structures of every infinite cardinality which are not secon...77%

    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