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    The Cantor-Lawvere principle (CL) is a theorem of first-o... — Carmelics
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    Home/Modality & Possibility
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    The Cantor-Lawvere principle (CL) is a theorem of first-order logic, not dependent on set-theoretic axioms

    Modality & PossibilityTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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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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    Reasons Against

    2 perspectives
    Reason against 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 against 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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    Related

    First-order logic itself presupposes a domain of quantification, which requires ...Lawvere's original 1969 formulation of the fixed-point theorem was explicitly ca...Quine and Boolos both argue that ontological commitments are smuggled into logic...Separating CL from set-theoretic axioms conflates syntactic derivability with se...
    +3 moreShow less
    The move from 'no set-theoretic axioms are cited' to 'no set-theoretic commitmen...The proof of CL does not rely essentially on any axiom of set theoryThe theorem asserting the non-existence of the set of non-self-membered sets sim...

    Similar

    CL applies to functions and is a principle of first-order logic86%New Rational Reflection is a Non-Akrasia principle85%Lindström (1969) proved that first-order logic is the strongest logic ...81%The completeness theorem for first-order logic can be obtained from an...80%

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    It’s worth mentioning, firstly, that this principle underlies diagonal argumentation in general (cf. Gaifman 2006). Even venerable examples such as Post’s informal argument that there is a recursively enumerable set of positive integers whose complement is not recursively enumerable, rely in essence on CL (Davis 1965: 312). A slight variant of CL is frequently found in the literature on undecidability (cf. Shoenfield 1967: 131). Secondly, the proof of CL does not rely essentially on any axiom
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    claim
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    3 (1 for, 2 against)
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