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    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Infinity is a theorem of NF rather than an axiom

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 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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      Think about whether this reason is strong or weak

    • 2.A theorem derived within a potentially inconsistent system carries no stronger modal force than an explicit axiom, since ex contradictione quodlibet trivializes both.
      ?

      Think about whether this reason is strong or weak

    • 3.The epistemological distinction between 'theorem' and 'axiom' collapses when the foundational system's consistency is not independently established.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    • 3.Poincaré's impredicativity objection applies here: proofs of infinity within a type-structured system tacitly presuppose the infinite domain they purport to derive.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.If the universe of NF were finite, it could be well-ordered
      ?

      Think about whether this reason is strong or weak

    • 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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    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.