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    ZFC set theory cannot serve as a sufficient basis for the... — Carmelics
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    ZFC set theory cannot serve as a sufficient basis for the mathematics of infinity.

    Skepticism
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Related

    A foundation is sufficient if it decides all practically significant infinitary ...A theory that is demonstrably incomplete with respect to important statements ab...Gödel and Cohen demonstrated the mathematical incompleteness of ZFC set theory w...Gödel himself regarded ZFC as capturing the iterative conception of set, which g...
    +3 moreShow less
    Incompleteness with respect to specific statements like CH does not entail insuf...Penelope Maddy and others in the set-theoretic naturalism tradition argue that Z...The independence of CH from ZFC reveals a genuine branching of mathematical poss...

    Similar

    A theory that is demonstrably incomplete with respect to important sta...89%Standard set theories entail that there actually exist huge infinities...83%ZFC set theory is demonstrably incomplete and cannot settle major ques...80%There is no such thing as an infinite mathematical domain (i.e., total...79%

    Source

    AI-extracted1/3 agreementValid
    SEP: infinity
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    Such results show that even an arithmetical theory such as PA can express statements of mathematical significance (as opposed to statements concocted for logical purposes) that require some detour through the infinite to be proved, even though they can be stated purely arithmetically. In contrast to arithmetic, the mathematical incompleteness of set theory was shown by Gödel and Cohen for important statements such as the Axiom of Choice, the continuum hypothesis, etc. It is important to emphasiz
    Extraction notes

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    Details

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