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    A system's inability to assign a cardinal to a totality w... — Carmelics
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    Challenges→In ZF and ZFC, the totality of transfinite cardinal numbers does not qualify as a set having a definite cardinal number of members.

    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.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.Formal systems are inherently limited tools; their incompleteness says nothing about mind-independent mathematical reality.
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    • 2.Cantor's transfinite cardinals exist beyond ZFC; some totalities have determinate sizes even if unprovable within particular axioms.
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    • 3.Physical infinities (universe, spacetime) plausibly have objective magnitudes regardless of what any formal system can express.
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    Reasons Against

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    Reason against
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    • 1.Without axioms defining 'size,' claiming a totality has 'determinate size' is metaphysically empty—size has no meaning independent of formal structure.
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    • 2.Appeals to ontological realism about inaccessible infinities lack empirical constraint; they're unfalsifiable metaphysical speculation.
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    • 3.If two axiom systems assign different cardinalities to the same totality, invoking mind-independent size resolves nothing—contradiction remains.
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    Related

    Appeals to ontological realism about inaccessible infinities lack empirical cons...Cantor's transfinite cardinals exist beyond ZFC; some totalities have determinat...Formal systems are inherently limited tools; their incompleteness says nothing a...If two axiom systems assign different cardinalities to the same totality, invoki...
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    In ZF and ZFC, the totality of transfinite cardinal numbers does not qualify as ...Physical infinities (universe, spacetime) plausibly have objective magnitudes re...Without axioms defining 'size,' claiming a totality has 'determinate size' is me...

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