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    The formula x = a ∨ x = b implicitly quantifies over the ... — Carmelics
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    Challenges→The Axiom of Pairing is true for sets

    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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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Predicativism requires that sets be definable without quantifying over larger domains containing the set itself.
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    • 2.The formula x = a ∨ x = b quantifies over all possible values of x to determine membership in {a,b}.
      ?

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    • 3.If {a,b} exists prior to this quantification, the quantifier presupposes the totality it purports to construct.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The disjunction x = a ∨ x = b is a finite, constructive definition requiring no quantification over infinite domains.
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    • 2.Predicativists permit impredicative operations on finite sets; circularity only arises with infinite totalities.
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    • 3.The formula defines {a,b} relative to a and b as given parameters, not by quantifying over all possible sets.
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    All sources support it1 linkedProof of definition segments1 linked

    Related

    If {a,b} exists prior to this quantification, the quantifier presupposes the tot...Predicativism requires that sets be definable without quantifying over larger do...Predicativists permit impredicative operations on finite sets; circularity only ...The Axiom of Pairing is true for sets
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    The disjunction x = a ∨ x = b is a finite, constructive definition requiring no ...The formula defines {a,b} relative to a and b as given parameters, not by quanti...The formula x = a ∨ x = b quantifies over all possible values of x to determine ...

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