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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that The equivalence of Hintikka's game-theoretic definition of truth and Tarski's definition of truth is itself equivalent to the axiom of choice (given the other axioms of Zermelo-Fraenkel set theory).

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Hodges (1985) demonstrated that the equivalence holds for first-order logic over well-orderable domains without invoking full AC.
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    • 2.The claim conflates the set-theoretic machinery needed for arbitrary structures with the logical equivalence itself, which is domain-relative.
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    • 3.Restricting quantifier domains to sets admitting a choice function is sufficient for GTS-Tarski equivalence in standard model-theoretic practice.
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    Reason for 2 of 2
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    • 1.Constructive and intuitionistic reformulations of GTS by Felscher (1985) achieve Tarski-equivalence without classical AC by replacing winning strategies with computable functions.
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    • 2.If a constructively valid proof of equivalence exists that avoids AC, then AC cannot be strictly necessary for the equivalence, only for one classical formulation of it.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.The argument that player ∃'s winning strategies for G(∀x φ(x)) can be assembled from strategies for G(φ(a)) requires the axiom of choice.
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    • 2.The axiom of choice is necessary and sufficient for the equivalence between game-theoretic and Tarskian truth definitions within ZF set theory.
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