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    If 'Zermelo's axioms' denotes Z rather than ZF, some inde... — Carmelics
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    Challenges→Zermelo's axioms for set theory are insufficient to prove the Axiom of Choice, or even that every infinite set is the disjoint union of two infinite sets.

    If 'Zermelo's axioms' denotes Z rather than ZF, some independence results require separate proofs, since Z and ZF have different proof-theoretic strengths.

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

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Z lacks the Axiom of Replacement, which is essential for constructing hierarchies beyond V_ω, affecting transfinite induction proofs.
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    • 2.Independence results like CH require different proof strategies in Z versus ZF due to differing proof-theoretic ordinals (ω_1^CK vs higher).
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    • 3.Confusing Z and ZF in mathematical literature creates pedagogical problems, since results proven in ZF may be unprovable or require modification in Z.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Most classical independence results (CH, AC independence) hold in both Z and ZF since they operate within comparable constructible hierarchies.
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    • 2.The historical convention names 'Zermelo's axioms' as ZF (including Replacement), making the claim's conditional premise ambiguous or non-standard.
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    • 3.Practical mathematics rarely depends on Z-specific limitations; independence proofs that differ between systems are specialized technical concerns, not central theorems.
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    Related

    Confusing Z and ZF in mathematical literature creates pedagogical problems, sinc...Independence results like CH require different proof strategies in Z versus ZF d...Most classical independence results (CH, AC independence) hold in both Z and ZF ...Practical mathematics rarely depends on Z-specific limitations; independence pro...
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    The historical convention names 'Zermelo's axioms' as ZF (including Replacement)...Z lacks the Axiom of Replacement, which is essential for constructing hierarchie...Zermelo's axioms for set theory are insufficient to prove the Axiom of Choice, o...

    Details

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