Meaning of urelement | Babel Free

Definitions

A mathematical object which is not a set but which can be an element of a set.

Equivalents

Examples

“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.”

“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, a, we can build⟳ the sets #92;empty and #92;#123;a#92;#125;. If there are two urelements, a and b, the possible sets are #92;#123;a#92;#125;, #92;#123;b#92;#125;, and #92;#123;a,b#92;#125;; 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⟳ #92;empty and #92;#123;#92;empty#92;#125;.”

“Let⟳ #92;mathcal#123;L#125;#92;alpha#95;#92;in be a version of the language of set-theory with urelements in which each occurrence of a quantifier is restricted by some V#95;#92;beta (#92;betalt;#92;alpha).”

CEFR level