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
    The Lévy hierarchy Σ_n ∪ Π_n in set theory does not colla... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    The Lévy hierarchy Σ_n ∪ Π_n in set theory does not collapse at the first level

    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.There is no known method to reduce the truth of an arbitrary set-theoretic formula to a Σ_1 ∪ Π_1 formula
      ?

      Think about whether this reason is strong or weak

    • 2.The decision problem for Σ_n ∪ Π_n formulas becomes strictly more complex as n increases
      ?

      Think about whether this reason is strong or weak

    • 3.The Löwenheim-Skolem and Hanf numbers for Σ_n ∪ Π_n in set theory grow strictly with n (Väänänen 1979)
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The independence of certain set-theoretic statements from ZFC (Cohen 1963) means hierarchy non-collapse may itself be a model-relative rather than absolute fact.
      ?

      Think about whether this reason is strong or weak

    • 2.If the non-collapse of Σ_n/Π_n levels depends on consistency assumptions unprovable in ZFC, the claim's necessity is undermined by Gödel's incompleteness theorems.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Putnam's model-theoretic argument shows that formal hierarchies underdetermine their intended interpretation, so complexity distinctions may reflect metatheoretic choices, not intrinsic ontological stratification.
      ?

      Think about whether this reason is strong or weak

    • 2.The Löwenheim-Skolem-based evidence (P3) establishes cardinality distinctions between models, not that the formulas themselves are irreducibly non-equivalent across all interpretations.
      ?

      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

    Connections

    1 topic

    Skepticism1 linked

    Related

    If the non-collapse of Σ_n/Π_n levels depends on consistency assumptions unprova...Putnam's model-theoretic argument shows that formal hierarchies underdetermine t...The Löwenheim-Skolem and Hanf numbers for Σ_n ∪ Π_n in set theory grow strictly ...The Löwenheim-Skolem-based evidence (P3) establishes cardinality distinctions be...
    +3 moreShow less
    The decision problem for Σ_n ∪ Π_n formulas becomes strictly more complex as n i...The independence of certain set-theoretic statements from ZFC (Cohen 1963) means...There is no known method to reduce the truth of an arbitrary set-theoretic formu...

    Similar

    The expressive hierarchy Σ^1_n ∪ Π^1_n of second-order logic already h...79%The Polynomial Hierarchy collapses if either the Buss hierarchy of the...77%If either the hierarchy of theories S^i_2, T^i_2 collapses or S_2 or T...77%The Polynomial Hierarchy does not collapse to any finite level (Σ^P_k ...76%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
    View source passageHide passage
    This means that the Löwenheim number[4] and the Hanf number[5] of the entire second-order logic are the same as those of the fragment \(\Pi^1_1\). Summing up, upon first inspection the levels \(\Sigma^1_n\) and \(\Pi^1_n\) of the hierarchy of second-order formulas grow strictly in expressive power as n increases, but a more careful analysis reveals that already the first level \(\Sigma^1_1\cup \Pi^1_1\) has the power of all the levels \(\Sigma^1_n, \Pi^1_n\) even if the power is somewhat im
    Extraction notes

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

    Details

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