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    If cardinal representation requires AC as a necessary con... — Carmelics
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    Challenges→Every cardinal number can be represented by an ordinal number

    If cardinal representation requires AC as a necessary condition, the claim is not a theorem of ZF but a conditional on a disputed axiom.

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

    Reasons For

    1 perspective
    Reason for
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    • 1.AC is independent of ZF: theorems requiring AC cannot be proven in ZF alone, making them contingent rather than foundational truths.
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    • 2.Mathematical claims should be classified by their axiomatic dependencies to avoid misleading practitioners about what actually follows from core principles.
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    • 3.Disputed axioms introduce non-standard models where cardinal representation may fail, so calling it a theorem obscures this mathematical reality.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Many 'ZF theorems' actually depend on hidden AC-like principles (replacement, infinity); the distinction between conditional and foundational is unclear.
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    • 2.Labeling AC-dependent results as 'conditionals' rather than theorems creates awkward notation and doesn't reflect how working mathematicians actually reason about cardinality.
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    • 3.If cardinal representation is central to mathematics, pragmatically treating it as a theorem (accepting AC) may be more useful than pedantic axiomatic tracking.
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    Connections

    1 linked claim · 1 topic

    Modality & Possibility1 linked
    Every cardinal number can be represented by an ordinal number

    Related

    AC is independent of ZF: theorems requiring AC cannot be proven in ZF alone, mak...Disputed axioms introduce non-standard models where cardinal representation may ...Every cardinal number can be represented by an ordinal numberIf cardinal representation is central to mathematics, pragmatically treating it ...
    +3 moreShow less
    Labeling AC-dependent results as 'conditionals' rather than theorems creates awk...Many 'ZF theorems' actually depend on hidden AC-like principles (replacement, in...Mathematical claims should be classified by their axiomatic dependencies to avoi...

    Details

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