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    The totality of transfinite cardinal numbers is absolutel... — Carmelics
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    The totality of transfinite cardinal numbers is absolutely infinite in a sui generis non-arithmetical sense, rather than having a cardinal number of its own.

    Divine Attributes
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.There is no largest transfinite cardinal number.
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    • 2.Although there are infinitely many transfinite cardinal numbers, there is no cardinal number of them.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Cantor's own distinction between the 'Absolute Infinite' and transfinite cardinals relies on a theological notion (the Absolute as God's mind) that does not survive secularization.
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    • 2.Without theological grounding, the claim that the totality is 'sui generis' collapses into either a set-theoretic proper class or an ad hoc stipulation.
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    • 3.Modern set theories like ZFC and NBG handle the totality of cardinals via the proper class Ord without invoking a non-arithmetical 'absolute infinity' as a distinct metaphysical category.
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    Reason against 2 of 2
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    • 1.If 'absolutely infinite' means having no cardinal number, this is precisely what ZFC already says about proper classes, making the 'sui generis' qualifier redundant rather than explanatory.
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    • 2.Claiming the totality is non-arithmetical in a unique sense smuggles in an unexplained primitive that violates the principle of ontological parsimony (Ockham's razor) in mathematical ontology.
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    Related

    Although there are infinitely many transfinite cardinal numbers, there is no car...Cantor's own distinction between the 'Absolute Infinite' and transfinite cardina...Claiming the totality is non-arithmetical in a unique sense smuggles in an unexp...If 'absolutely infinite' means having no cardinal number, this is precisely what...
    +3 moreShow less
    Modern set theories like ZFC and NBG handle the totality of cardinals via the pr...There is no largest transfinite cardinal number.Without theological grounding, the claim that the totality is 'sui generis' coll...

    Similar

    Although there are infinitely many transfinite cardinal numbers, there...90%In ZF and ZFC, the totality of transfinite cardinal numbers does not q...89%There is no largest transfinite cardinal number.84%There is no proportion between the infinite and the finite.74%

    Source

    AI-extracted2/3 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 these premises and draws this conclusion as part of Cantor's view: that there is no largest transfinite cardinal, that there are infinitely many but no cardinal number of them, and that the totality is therefore absolutely infinite in a sui generis non-arithmetical sense.

    Confidence: Clearly attributed to Cantor's reasoning in the passage.

    Details

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