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    Made withinDC&Austin
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    321,452
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    42
    The Axiom of Pairing is true for sets — Carmelics
    Home/Proof of definition segments
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    The Axiom of Pairing is true for sets

    All sources support itProof of definition segments
    ?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 formula x = a ∨ x = b (where a and b are sets) does not mention sethood
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    • 2.The formula has only the sets a and b as parameters
      ?

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    • 3.The formula is true only of sets
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.P4 assumes a separation-style comprehension schema, but this presupposes a background domain of sets that is itself not justified by the argument.
      ?

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    • 2.The Axiom of Pairing is not derivable from comprehension alone without prior commitment to the existence of a set-forming operation, making P4 question-begging.
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    Reason against 2 of 2
    ?
    • 1.Predicativist critics like Poincaré and Weyl hold that definitions quantifying over all sets are impredicative and cannot ground new set existence claims.
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    • 2.The formula x = a ∨ x = b implicitly quantifies over the totality of sets to isolate {a,b}, making its use in P4 viciously circular on predicativist grounds.
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    Topics

    Proof of definition segmentsAll sources support it

    Connections

    2 topics

    Philosophy of Language2 linkedTruth & Knowledge1 linked

    Related

    Any formula that does not mention sethood, has only sets as parameters, and is t...P4 assumes a separation-style comprehension schema, but this presupposes a backg...Predicativist critics like Poincaré and Weyl hold that definitions quantifying o...The Axiom of Pairing is not derivable from comprehension alone without prior com...
    +4 moreShow less
    The formula has only the sets a and b as parametersThe formula is true only of setsThe formula x = a ∨ x = b (where a and b are sets) does not mention sethoodThe formula x = a ∨ x = b implicitly quantifies over the totality of sets to iso...

    Similar

    The formula is true only of sets90%The Power Set axiom is true for sets.88%The Union axiom is true for sets86%The formula ∀y(y ∈ x → y ∈ a) is true only of sets by the Axiom of Sub...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
    View source passageHide passage
    The formula \(x = a \lor x = b\) (where \(a\) and \(b\) are sets) does not mention sethood, has only the sets \(a\) and \(b\) as parameters, and is true only of sets. Thus it defines a set, and Pairing is true for sets.
    Extraction notes

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

    Details

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