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    The Axiom of Limitation of Size implies the Axiom of Choice — Carmelics
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    The Axiom of Limitation of Size implies the Axiom of Choice

    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.The von Neumann ordinals form a proper class, essentially by the Burali-Forti paradox
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    • 2.Because the von Neumann ordinals form a proper class, the universe must be the same size as the class of ordinals
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    • 3.A class bijection between the universe and the ordinals can be used to define a global well-ordering of the universe
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The derivation assumes the Axiom of Replacement to establish that the von Neumann ordinals constitute a proper class via Burali-Forti.
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    • 2.The Axiom of Limitation of Size, as formulated by von Neumann, is itself equivalent to Replacement plus Choice, making the implication circular.
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    • 3.Cantor-von Neumann set theories that restrict Replacement can model Limitation of Size without generating a class-sized ordinal spine sufficient to well-order the universe.
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    Reason against 2 of 2
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    • 1.Michael Hallett and Akihiro Kanamori argue that 'same size' for proper classes requires a prior notion of class cardinality that is not definable without Choice-like assumptions.
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    • 2.If class bijections require definable functions, the step from equinumerosity with the ordinals to a global well-ordering presupposes definable choice functions, making the argument question-begging.
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    Related

    A class bijection between the universe and the ordinals can be used to define a ...Because the von Neumann ordinals form a proper class, the universe must be the s...Cantor-von Neumann set theories that restrict Replacement can model Limitation o...If class bijections require definable functions, the step from equinumerosity wi...
    +5 moreShow less
    Michael Hallett and Akihiro Kanamori argue that 'same size' for proper classes r...The Axiom of Limitation of Size, as formulated by von Neumann, is itself equival...The derivation assumes the Axiom of Replacement to establish that the von Neuman...The existence of a global well-ordering of the universe immediately implies the ...The von Neumann ordinals form a proper class, essentially by the Burali-Forti pa...

    Similar

    The Axiom of Limitation of Size implies the Axiom of Replacement100%The Manual of Reason provides its own definition of yogyatā86%Under the Piece-of-Pie Model, there is a part of the Form of the F in ...84%Therefore, F ≠ G implies the extension of F ≠ the extension of G.77%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
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    This elegant axiom is essentially due to von Neumann. A class bijection is a class of ordered pairs; there might be pathology here if we did not have enough pairs as sets, but other axioms do provide for their existence. It is interesting to observe that this axiom implies Replacement (a class which is the same size as a set cannot be the same size as the universe) and, surprisingly, implies Choice (the von Neumann ordinals make up a proper class essentially by the Burali-Forti paradox, so the u
    Extraction notes

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

    Details

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