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    The Axiom of Choice is innocent in the context of second-... — Carmelics
    Home/Philosophy of Language
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    The Axiom of Choice is innocent in the context of second-order logic and is generally accepted in the second-order logic literature.

    Philosophy of LanguageTruth & 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 basic tenet of second-order logic is that all properties of elements of a fixed domain exist and can be quantified over.
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    • 2.In set-theoretical terms, all subsets—definable or not—of a set of any cardinality exist and can be quantified over.
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    • 3.Quantifying over all subsets naturally licenses the Axiom of Choice for families of subsets of any cardinality.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Second-order logic's quantification over all subsets is a semantic stipulation, not an ontological guarantee that choice functions exist.
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    • 2.The Axiom of Choice asserts the existence of a specific function, which requires more than mere acknowledgment that subsets can be quantified over.
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    • 3.Boolos and others have shown that the 'full' second-order semantics presupposes a background set theory that itself requires AC, making the 'innocence' claim circular.
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    Reason against 2 of 2
    ?
    • 1.Constructivist and predicativist traditions (Weyl, Feferman) reject impredicative quantification over all subsets as ontologically illegitimate.
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    • 2.If second-order quantification is restricted to predicatively definable properties, as Feferman's program demands, AC is not automatically licensed.
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    Topics

    Philosophy of LanguageTruth & Knowledge

    Connections

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    Modality & Possibility3 linked

    Related

    Boolos and others have shown that the 'full' second-order semantics presupposes ...Constructivist and predicativist traditions (Weyl, Feferman) reject impredicativ...If second-order quantification is restricted to predicatively definable properti...In set-theoretical terms, all subsets—definable or not—of a set of any cardinali...
    +4 moreShow less
    Quantifying over all subsets naturally licenses the Axiom of Choice for families...Second-order logic's quantification over all subsets is a semantic stipulation, ...The Axiom of Choice asserts the existence of a specific function, which requires...The basic tenet of second-order logic is that all properties of elements of a fi...

    Similar

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    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    Let \(\theta_{\le}(P,R)\) be the formula \[ \exists F\left(\forall x\,\forall y\left( (F(x)=F(y)\to x=y) \land(P(x)\to R(F(x)) \right)\right). \] Now \(\mm\models_s\theta_\le(P,R)\) if and only if \(|s(P)|\le |s(R)|\). Let \(\theta_{\textrm{EQ}}(P,R)\) be the formula \(\theta_{{\le}}(P,R)\land \theta_{{\le}}(R,P)\). Now \(\mm\models_s\phi(P,R)\) if and only if \(|s(P)|=|s(R)|\). Let \(\theta'_{\textrm{EC}}(Y)\) be \[ \exists F\left( \forall x\,\forall y((F(x)=F(y)\to x=y)\land R(F(x)))
    Extraction notes

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

    Details

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