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    Zermelo's axioms for set theory are insufficient to prove... — Carmelics
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    Home/Modality & Possibility
    HistoryEditSee Inverse

    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.

    Modality & PossibilityTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Related

    If 'Zermelo's axioms' denotes Z rather than ZF, some independence results requir...Lindenbaum and Mostowski (1938) established this result using the Fraenkel-Mosto...The Fraenkel-Mostowski construction produces a Henkin (set-theoretic) model in w...The Fraenkel-Mostowski method applies to set theory with atoms (ZFA), not standa...
    +4 moreShow less
    The claim conflates the historical Zermelo system with the modern ZF framework, ...Therefore, the FM construction alone does not establish the claim about Zermelo'...Transferring independence results from ZFA to ZF requires the additional Jech-So...Zermelo's 1908 axioms are not identical to ZF; they lack the Axiom of Replacemen...

    Similar

    As in set theory without the Axiom of Choice, distinct characterizatio...85%ZFC set theory cannot serve as a sufficient basis for the mathematics ...79%P is a necessarily infinite mutually disjoint set of pairs partitionin...79%Banks et al. (1992) proved that conditions (2) and (3) imply condition...77%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    For more on the proof theory of second-order logic, see Buss (1998). There is a translation of many sorted logic further to single sorted first order logic due essentially to Herbrand (1930), see also Wang (1952) and Schmidt (1951). This can be used to obtain many of the basic properties of first order logic first for many sorted logic and then further for second-order logic with general models. The most important application of general models is the Completeness Theorem: Theorem 14 (Henkin 1
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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