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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
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    42
    Home/Original/inverse
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    Inverse View

    It is not the case that 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.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The Fraenkel-Mostowski method applies to set theory with atoms (ZFA), not standard Zermelo-Fraenkel set theory, which lacks urelements.
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    • 2.Transferring independence results from ZFA to ZF requires the additional Jech-Sochor embedding theorem, which is a non-trivial further result.
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    • 3.Therefore, the FM construction alone does not establish the claim about Zermelo's axioms (ZF) without invoking machinery beyond what the argument cites.
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    Reason for 2 of 2
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    • 1.Zermelo's 1908 axioms are not identical to ZF; they lack the Axiom of Replacement, making the target system ambiguous in independence proofs.
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    • 2.If 'Zermelo's axioms' denotes Z rather than ZF, some independence results require separate proofs, since Z and ZF have different proof-theoretic strengths.
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    • 3.The claim conflates the historical Zermelo system with the modern ZF framework, undermining the precision required for the metalogical conclusion.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Lindenbaum and Mostowski (1938) established this result using the Fraenkel-Mostowski method.
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    • 2.The Fraenkel-Mostowski construction produces a Henkin (set-theoretic) model in which the Axiom of Choice fails and the relevant partition property fails.
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