Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

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

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Small large cardinal axioms are compatible with V=L, unli... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    Small large cardinal axioms are compatible with V=L, unlike measurable cardinals

    Modality & Possibility
    ?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.For each small large cardinal axiom φ, if ZFC+φ is consistent then ZFC+φ+V=L is also consistent
      ?

      Think about whether this reason is strong or weak

    • 2.ZFC + 'There is a measurable cardinal' proves ¬V=L, making measurable cardinals incompatible with V=L
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The consistency of ZFC+φ+V=L establishes only proof-theoretic compatibility, not genuine ontological cohabitation in any intended model.
      ?

      Think about whether this reason is strong or weak

    • 2.Scott's 1961 theorem shows measurable cardinals destroy V=L by producing non-constructible sets, revealing that 'compatibility' conceals a deeper modal distinction between what can be consistently asserted and what can actually obtain simultaneously.
      ?

      Think about whether this reason is strong or weak

    • 3.A philosophically robust notion of compatibility requires satisfaction in a single intended model, not merely in separate formal systems that share no common universe.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.The boundary between 'small' and 'large' large cardinals is itself set-theoretically defined, making the claim epistemically circular when used to ground a philosophical distinction about possibility.
      ?

      Think about whether this reason is strong or weak

    • 2.Woodin's work on the inner model program suggests that sufficiently strong small large cardinals may eventually yield incompatibility with V=L, undermining the stability of the claimed dichotomy.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

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

    Topics

    Modality & PossibilityTruth & Knowledge

    Related

    A philosophically robust notion of compatibility requires satisfaction in a sing...For each small large cardinal axiom φ, if ZFC+φ is consistent then ZFC+φ+V=L is ...Scott's 1961 theorem shows measurable cardinals destroy V=L by producing non-con...The boundary between 'small' and 'large' large cardinals is itself set-theoretic...
    +3 moreShow less
    The consistency of ZFC+φ+V=L establishes only proof-theoretic compatibility, not...Woodin's work on the inner model program suggests that sufficiently strong small...ZFC + 'There is a measurable cardinal' proves ¬V=L, making measurable cardinals ...

    Similar

    ZFC + 'There is a measurable cardinal' proves ¬V=L, making measurable ...84%Scott proved this result in contrast to small large cardinals, which a...84%For each small large cardinal axiom φ, if ZFC+φ is consistent then ZFC...83%The theory ZFC + 'There is a measurable cardinal' proves ¬V=L80%

    Source

    AI-extracted1/3 agreementValid
    SEP: independence-large-cardinals
    View source passageHide passage
    In fact, Scott showed that (in contrast to the small large cardinals) measurable cardinals cannot exist in Gödel's constructible universe. Let us be precise about this. Let V=L be the statement that asserts that all sets are constructible. Then for each small large cardinal axiom φ (to be precise, those listed above) if the theory ZFC+φ is consistent then so is the theory ZFC+φ+V=L. In contrast, the theory ZFC + “There is a measurable cardinal” proves ¬V=L. This may seem somewhat counterintu
    Extraction notes

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

    Details

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