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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that ZFC set theory cannot serve as a sufficient basis for the mathematics of infinity.

    ?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.Incompleteness with respect to specific statements like CH does not entail insufficiency for the bulk of mathematical practice concerning infinity.
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    • 2.Gödel himself regarded ZFC as capturing the iterative conception of set, which grounds transfinite arithmetic, cardinal arithmetic, and ordinal theory robustly.
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    • 3.A foundation is sufficient if it decides all practically significant infinitary questions, even if it leaves certain independence results undecided.
      ?

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    Reason for 2 of 2
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    • 1.The independence of CH from ZFC reveals a genuine branching of mathematical possibility, not a defect in ZFC's foundational adequacy.
      ?

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    • 2.Penelope Maddy and others in the set-theoretic naturalism tradition argue that ZFC's open-endedness is a feature reflecting the richness of the infinite, not a failure of sufficiency.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Gödel and Cohen demonstrated the mathematical incompleteness of ZFC set theory with respect to important statements such as the Axiom of Choice and the Continuum Hypothesis.
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    • 2.A theory that is demonstrably incomplete with respect to important statements about infinity cannot be taken as a sufficient basis for the mathematics of infinity.
      ?

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