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 apparent bijection x ↦ {x} between ℘₁(V) and V cannot... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    The apparent bijection x ↦ {x} between ℘₁(V) and V cannot be a set in NFU.

    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.The map x ↦ {x} has an unstratified definition.
      ?

      Think about whether this reason is strong or weak

    • 2.In NFU, only stratified definitions give rise to sets.
      ?

      Think about whether this reason is strong or weak

    • 3.There is no expectation that an unstratified definition yields a set.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Stratification is a syntactic criterion, but the existence of a function as a set is a semantic question about the model's domain.
      ?

      Think about whether this reason is strong or weak

    • 2.Holmes and others have shown NFU admits non-standard models where some unstratified operations have set-realizations under alternative interpretations.
      ?

      Think about whether this reason is strong or weak

    • 3.The failure of x↦{x} to be a set is a feature of standard NFU models, not a logical necessity derivable from stratification alone.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Quine's original NF and Jensen's NFU differ in that NFU's urelements create asymmetry: V and ℘₁(V) can have different cardinalities, making the bijection's non-existence a cardinality fact, not merely a stratification fact.
      ?

      Think about whether this reason is strong or weak

    • 2.Presenting stratification failure as the explanation conflates the proof-theoretic reason for unprovability with the model-theoretic reason for the bijection's non-existence as a set.
      ?

      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 & PossibilityProof of definition segments

    Connections

    1 topic

    Truth & Knowledge3 linked

    Related

    Holmes and others have shown NFU admits non-standard models where some unstratif...In NFU, only stratified definitions give rise to sets.Presenting stratification failure as the explanation conflates the proof-theoret...Quine's original NF and Jensen's NFU differ in that NFU's urelements create asym...
    +4 moreShow less
    Stratification is a syntactic criterion, but the existence of a function as a se...The failure of x↦{x} to be a set is a feature of standard NFU models, not a logi...The map x ↦ {x} has an unstratified definition.There is no expectation that an unstratified definition yields a set.

    Similar

    If a bijection exists between two sets, those sets have the same cardi...77%The formula ∀y(y ∈ x → y ∈ a) is true only of sets by the Axiom of Sub...75%A set M with the property that KS1 and KS2 are contradictory exists75%Because MΓ∪{A} is not assumed to be a subset of MΓ, the implication fr...75%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
    View source passageHide passage
    The Cantor theorem of the usual set theory asserts that \(|A| \lt |\wp(A)|\). This is clearly not true in NFU, since | \(V|\) is the cardinality of the universe and \(|\wp(V)|\) is the cardinality of the set of sets, and in fact \(|V| \gt \gt |\wp(V)|\) in all known models of NFU (there are many intervening cardinals in all such models). But \(|A| \lt |\wp(A)|\) does not make sense in TST: it is ill-typed. The correct theorem in TST, which is inherited by NFU, is \(|\wp_1 (A)| \lt |\wp(A)|\), wh
    Extraction notes

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

    Details

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