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    In ZF and ZFC, the totality of transfinite cardinal numbe... — Carmelics
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    In ZF and ZFC, the totality of transfinite cardinal numbers does not qualify as a set having a definite cardinal number of members.

    Divine Attributes
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.Such a set would be too large.
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    • 2.Were such a set to exist, paradoxical consequences would ensue akin to Russell's paradox.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Cantor himself distinguished between 'consistent multiplicities' (sets) and 'inconsistent multiplicities' (absolute infinities), treating the latter as mathematically real but beyond formal set membership.
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    • 2.The claim conflates a formal ZF/ZFC limitation with a metaphysical impossibility, when Cantor's absolute infinite (Ω) was intended as a positive theological-mathematical concept, not merely a prohibited construction.
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    • 3.A system's inability to assign a cardinal to a totality within its own axioms does not entail that the totality lacks a determinate size in any stronger ontological sense.
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    Reason against 2 of 2
    ?
    • 1.Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set theory formally accommodate proper classes, including the class of all cardinals, as legitimate mathematical objects with determinate extensions.
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    • 2.If the claim's force depends specifically on ZF/ZFC axiomatics, it is a contingent artifact of one foundational choice rather than a necessary truth about cardinality itself, undermining its use in broader theological arguments about divine power.
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    Related

    A system's inability to assign a cardinal to a totality within its own axioms do...Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set the...Cantor himself distinguished between 'consistent multiplicities' (sets) and 'inc...If the claim's force depends specifically on ZF/ZFC axiomatics, it is a continge...
    +3 moreShow less
    Such a set would be too large.The claim conflates a formal ZF/ZFC limitation with a metaphysical impossibility...Were such a set to exist, paradoxical consequences would ensue akin to Russell's...

    Similar

    The totality of transfinite cardinal numbers is absolutely infinite in...89%Although there are infinitely many transfinite cardinal numbers, there...86%There is no largest transfinite cardinal number.83%The actually infinite set of natural numbers and each of its members a...73%

    Source

    AI-extracted3/3 agreementValid
    SEP: omnipotence
    View source passageHide passage
    The founder of transfinite arithmetic, Georg Cantor (1845–1918), is also a founding father of set theory. He famously proved that the set of real numbers has a larger cardinal number than the set of natural numbers; the set of reals has the same cardinality as the power set (the set of all subsets) of the set of naturals. Cantor further argued that \(\aleph_0\) is the first (and smallest) transfinite cardinal number in an infinite series of increasingly larger transfinite cardinals, \(\aleph_0,\) \(\aleph_1,\) \(\aleph_2\), and so on. But note that the numerical subscripts of these alephs do n...
    Extraction notes

    Validity: The passage explicitly states that the totality of transfinite cardinal numbers does not qualify as a set in ZF/ZFC, and directly provides both premises—that such a set would be "too large" and that "paradoxical consequences would ensue akin to Russell's paradox"—as the supporting reasons.

    Confidence: Explicit argumentative structure in the text.

    Details

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