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    Infinity is a theorem of NF rather than an axiom — Carmelics
    Home/Modality & Possibility
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    Infinity is a theorem of NF rather than an axiom

    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.If the universe of NF were finite, it could be well-ordered
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    • 2.The universe of NF cannot be well-ordered
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    • 3.Therefore the universe of NF cannot be finite
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The proof that the universe of NF cannot be well-ordered itself presupposes the consistency of NF, which remains unproven for decades after Quine's 1937 formulation.
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    • 2.A theorem derived within a potentially inconsistent system carries no stronger modal force than an explicit axiom, since ex contradictione quodlibet trivializes both.
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    • 3.The epistemological distinction between 'theorem' and 'axiom' collapses when the foundational system's consistency is not independently established.
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    Reason against 2 of 2
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    • 1.Holmes's 2024 consistency proof of NF relative to ZFC relies on complex machinery that shifts, rather than eliminates, the foundational burden onto ZFC's own infinite ontology.
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    • 2.Deriving infinity as a theorem by embedding its content into stratification constraints smuggles infinitary commitments into the logical syntax rather than genuinely avoiding them.
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    • 3.Poincaré's impredicativity objection applies here: proofs of infinity within a type-structured system tacitly presuppose the infinite domain they purport to derive.
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    Related

    A theorem derived within a potentially inconsistent system carries no stronger m...Deriving infinity as a theorem by embedding its content into stratification cons...Holmes's 2024 consistency proof of NF relative to ZFC relies on complex machiner...If the universe of NF were finite, it could be well-ordered
    +5 moreShow less
    Poincaré's impredicativity objection applies here: proofs of infinity within a t...The epistemological distinction between 'theorem' and 'axiom' collapses when the...The proof that the universe of NF cannot be well-ordered itself presupposes the ...The universe of NF cannot be well-orderedTherefore the universe of NF cannot be finite

    Similar

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    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
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    No contradictions are known to follow from NF, but some uncomfortable consequences do follow. The Axiom of Choice is known to fail in NF: Specker (1953) proved that the universe cannot be well-ordered. (Since the universe cannot be well-ordered, it follows that the “Axiom” of Infinity is a theorem of NF: if the universe were finite, it could be well-ordered.) This might be thought to be what one would expect on adopting such a dangerous comprehension scheme, but this turns out not to be the prob
    Extraction notes

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

    Details

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