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    Quine's original NF and Jensen's NFU differ in that NFU's... — Carmelics
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    Challenges→The apparent bijection x ↦ {x} between ℘₁(V) and V cannot be a set in NFU.

    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.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Urelements in NFU break the self-applicability that forces NF's cardinality constraints, allowing genuine set-theoretic distinctions absent in NF.
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    • 2.If V and ℘₁(V) have different cardinalities in NFU, this reflects fundamental ontological asymmetry rather than mere syntactic stratification artifacts.
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    • 3.NFU's consistency relative to ZFC suggests its cardinality facts track genuine mathematical reality, not arbitrary formal restrictions.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Stratification constraints in NFU still govern what bijections are *definable*, so cardinality differences may remain formal rather than deep.
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    • 2.Urelements' non-membership in any set doesn't obviously make cardinality gaps ontologically significant versus NF's elegant type-theoretic closure.
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    • 3.Both theories preserve bijection-non-existence; calling one 'cardinality fact' and the other 'stratification fact' may be terminological, not substantive.
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    Key Terms

    Bijection(in set theory and mathematics)
    A perfect one-to-one matching between two sets, where every element in one set pairs with exactly one element in the other, and nothing is left unpaired.
    Jensen(in mathematical logic)
    Ronald Jensen is a mathematical logician who developed important ideas in set theory, building on work by other mathematicians and philosophers.
    NF(Formal set theory)
    A set theory (New Foundations) with a stratified comprehension scheme, from which no contradictions are currently known to follow but which has uncomfortable consequences including the failure of the Axiom of Choice
    NFU(in set theory)
    A modified version of Quine's NF system that Jensen created; it adds extra objects called 'urelements' to make the system work more flexibly.
    Quine(as a proper name referring to the philosopher whose theory is being discussed)
    Willard Van Orman Quine was a 20th-century American philosopher who wrote about how we know things and how language works. In this statement, we're discussing one of his specific ideas about observation.
    Stratification(in formal logic and set theory)
    In Quine's logic, a property of formulas where you can assign levels or ranks to variables in a way that respects the rules of the system—essentially, a formal way of keeping things organized and consistent.
    Urelements(in set theory)
    Basic objects in set theory that aren't sets themselves—they're the fundamental building blocks, like atoms, that other sets are built from.
    cardinality(Central to comparing infinite sets and establishing that no universal set exists.)
    A measure of the size of a set, indicating the number of elements it contains.
    ℘₁(V)(in mathematical notation)
    A mathematical notation meaning 'the power set of V,' which is the collection of all possible subsets (smaller groups) that can be made from the set V.

    Connections

    2 topics

    Proof of definition segments1 linkedModality & Possibility1 linked

    Related

    Both theories preserve bijection-non-existence; calling one 'cardinality fact' a...If V and ℘₁(V) have different cardinalities in NFU, this reflects fundamental on...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    NFU's consistency relative to ZFC suggests its cardinality facts track genuine m...
    Stratification constraints in NFU still govern what bijections are *definable*, ...
    +3 moreShow less
    The apparent bijection x ↦ {x} between ℘₁(V) and V cannot be a set in NFU.Urelements in NFU break the self-applicability that forces NF's cardinality cons...Urelements' non-membership in any set doesn't obviously make cardinality gaps on...