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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that The Cantor-Lawvere principle (CL) is a theorem of first-order logic, not dependent on set-theoretic axioms

    ?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.First-order logic itself presupposes a domain of quantification, which requires implicit set-theoretic or type-theoretic commitments.
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    • 2.Lawvere's original 1969 formulation of the fixed-point theorem was explicitly categorical, relying on topos-theoretic structure beyond bare first-order logic.
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    • 3.Separating CL from set-theoretic axioms conflates syntactic derivability with semantic validity, since standard model theory for FOL employs sets.
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    Reason for 2 of 2
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    • 1.Quine and Boolos both argue that ontological commitments are smuggled into logical frameworks through comprehension-like principles implicit in predicate abstraction.
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    • 2.The move from 'no set-theoretic axioms are cited' to 'no set-theoretic commitments are made' commits the use-mention fallacy regarding background mathematical infrastructure.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The proof of CL does not rely essentially on any axiom of set theory
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    • 2.The theorem asserting the non-existence of the set of non-self-membered sets similarly requires no set-theoretic axioms
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