mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself...

property (like finiteness in a hereditarily finite set). Some authors regard a nested set collection as a family of sets. Others prefer to classify it...

hereditary set. Hereditarily countable set Hereditarily finite set Well-founded set Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs...

first-order set theory. A set is hereditarily countable if and only if it is countable, and every element of its transitive closure is countable. Hereditarily finite...

mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle...

is the set of natural numbers, then Vω is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity. Vω+ω is the...

hereditarily finite set is defined as a finite set consisting of zero or more hereditarily finite sets. Equivalently, a set is hereditarily finite if and only...

and principle Computable set Diaconescu's theorem Disjunction and existence properties Epsilon-induction Hereditarily finite set Heyting arithmetic Impredicativity...

Hiranandani Foundation Schools, in India Hemifacial spasm, in neurology Hereditarily finite set, in mathematics Hexafluorosilicic acid, in chemistry Hydrogen forward...

than numbers to do the encoding. In simple cases when one uses a hereditarily finite set to encode formulas this is essentially equivalent to the use of...

the set N of all natural numbers does not (although it is a subset of S{}). In fact, the superstructure over {} consists of all of the hereditarily finite...

the a set is hereditarily P if all elements of its transitive closure have property P. Examples: Hereditarily countable set Hereditarily finite set Hessenberg...

a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows...

of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set. The notation for...

V_{\omega }\!} , the class of hereditarily finite sets, with the inherited membership relation. Note that if the axiom of the empty set is not taken as a part...

Grothendieck universes (other than the empty set and the set V ω {\displaystyle V_{\omega }} of all hereditarily finite sets) is not implied by the usual ZF axioms;...

constructed non-randomly, by symmetrizing the membership relation of the hereditarily finite sets, by applying the BIT predicate to the binary representations of...

Kripke–Platek set theory (Barwise 1975). The smallest example of an admissible set is the set of hereditarily finite sets. Another example is the set of hereditarily...

basis, every vector space has a (possibly infinite) vector basis. Hereditarily finite set Jech, Thomas J. (2008) [1973]. The Axiom of Choice. Dover Publications...

{\displaystyle V_{\omega }} is the set of hereditarily finite sets, i.e. the first set in the Von Neumann universe of non-finite rank. The choice function is...

any finite set. And this could be used to generate all hereditarily finite sets without using the axiom of union. Together with the axiom of empty set and...

algorithm that computes the membership of every natural number in a finite number of steps. A set is noncomputable (or undecidable) if it is not computable. A...

or even other sets. A set may be finite or infinite, depending whether the number of its elements is finite or not. There is a unique set with no elements...

used notion of compactness, which requires the existence of a finite subcover. A hereditarily Lindelöf space is a topological space such that every subspace...

cardinality: Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a finite set. Any set X that has the same...

hereditarily finite sets, and defining the adjacency relation of the Rado graph. In computer science, it is used for efficient representations of set...

simple examples of Grothendieck universes: The empty set, and The set of all hereditarily finite sets V ω {\displaystyle V_{\omega }} . Other examples are...

A} of subsets of a set X {\displaystyle X} is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A...

topology induced by a total order may be shown to be hereditarily normal. A totally ordered set is said to be complete if every nonempty subset that has...

to functions f that can be represented as sets as opposed to undefinable classes. The hereditarily finite sets, Vω, satisfy the axiom of regularity (and...