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    NFSI is consistent — Carmelics
    Statements
    321,452
    Perspectives
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    Topics
    42
    Home/Modality & Possibility
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    NFSI is consistent

    Modality & PossibilityTruth & 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.A structure exists in which the sets are exactly the finite and cofinite collections and this structure satisfies NFSI
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    • 2.A theory is consistent if a model of it exists
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.By Gödel's second incompleteness theorem, any consistency proof of NFSI conducted within a system of comparable strength is epistemically circular.
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    • 2.The metatheory used to verify the finite/cofinite structure is a model must itself be stronger than NFSI, making the consistency claim relative rather than absolute.
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    • 3.Holmes's 2024 claimed proof of NF's consistency, on which NFSI consistency claims partly depend, remained under peer scrutiny and is not yet universally accepted.
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    Reason against 2 of 2
    ?
    • 1.The finite/cofinite structure satisfies only a fragment of NFSI, and full NFSI includes stratified comprehension axioms not modeled by that structure.
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    • 2.A consistency proof relative to a restricted model does not establish consistency of the full theory unless the model is demonstrably a model of all axioms.
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    Connections

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    Related

    A consistency proof relative to a restricted model does not establish consistenc...A structure exists in which the sets are exactly the finite and cofinite collect...A theory is consistent if a model of it existsBy Gödel's second incompleteness theorem, any consistency proof of NFSI conducte...
    +3 moreShow less
    Holmes's 2024 claimed proof of NF's consistency, on which NFSI consistency claim...The finite/cofinite structure satisfies only a fragment of NFSI, and full NFSI i...The metatheory used to verify the finite/cofinite structure is a model must itse...

    Similar

    Peano Arithmetic (PA) is consistent.94%F is assumed to be consistent93%Arithmetic is consistent (contains no contradictions).92%HA + MP + CT is consistent91%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
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    Very recently, Sergei Tupailo (2010) has proved the consistency of NFSI, the fragment of NF consisting of extensionality and those instances of Comprehension (\(\{x \in A \mid \phi \}\) exists) which are stratified and in which the variable \(x\) is assigned the lowest type. Tupailo’s proof is highly technical, but Marcel Crabbé pointed out that a structure for the language of set theory in which the sets are exactly the finite and cofinite collections satisfies this theory (so it is very weak).
    Extraction notes

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

    Details

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