urelement

• ur-element

From ur- (“primordial”) +‎ element.

urelement (plural urelements)

1. (set theory) A mathematical object which is not a set but which can be an element of a set.

   Synonym: atom

   • 1996, Scientific Books staff (translators), Yuri L. Ershov, Definability and Computability, Scientific Books, page viii,

     The introduction of urelements would seem to be a technical improvement; however, now we know that just such an extension of the notion of the admissible set led to the universal theory of computability based on the notion of definability by formulas with (in a broad sense) effective semantics.

   • 2012, Nicholas J. J. Smith, Logic: The Laws of Truth, Princeton University Press, page 448:

     There may be no urelements; as we shall see, we can still build plenty of sets in this case. At stage 0 we can always build the empty set. If there are no urelements, this is the only set we can build. If there is one urelement, ${\displaystyle a}$, we can build the sets ${\displaystyle \emptyset }$ and ${\displaystyle \{a\}}$. If there are two urelements, ${\displaystyle a}$ and ${\displaystyle b}$, the possible sets are ${\displaystyle \{a\}}$, ${\displaystyle \{b\}}$, and ${\displaystyle \{a,b\}}$; and so on if there are more urelements.
     At stage 1, we can build any set containing urelements or sets built at stage 0; that is, any set whose members are already available at the beginning of stage 1. If there are no urelements, we can build ${\displaystyle \emptyset }$ and ${\displaystyle \{\emptyset \}}$.

   • 2013, Agustín Rayo, The Construction of Logical Space, Oxford University Press, page 95:

     Let ${\displaystyle {\mathcal {L}}_{\in }^{\alpha }}$ be a version of the language of set-theory with urelements in which each occurrence of a quantifier is restricted by some ${\displaystyle V_{\beta }}$ ${\displaystyle (\beta <\alpha )}$.

The standard axiomatisation of set theory, ZFC, ignores urelements. (By the axiom of extensionality, two sets whose only difference is that one contains urelements which the other does not would be equal.)

• KPU
• NFU

• hereditary set
• pure set
• Quine atom

• Urelement on Wikipedia.Wikipedia
• Axiom of extensionality on Wikipedia.Wikipedia
• Hereditary set on Wikipedia.Wikipedia
• Kripke–Platek set theory with urelements on Wikipedia.Wikipedia
• New Foundations on Wikipedia.Wikipedia
• urelement on nLab
• Urelement on Wolfram MathWorld