In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. The inductive definition above is well-founded...

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...

mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is countable...

set theories in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set. Hereditarily countable set...

inheritance of object-oriented programming. Hereditarily countable set Hereditary property Hierarchy (mathematics) Nested set model for storing hierarchical information...

sets. Another example is the set of hereditarily countable sets. Admissible ordinal Barwise, Jon (1975). Admissible Sets and Structures: An Approach to Definability...

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

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence...

theorems in set theory, such as the Mostowski collapse lemma. Constructible universe Admissible ordinal Hereditarily countable set Kripke–Platek set theory...

A hereditarily countable set is a countable set of hereditarily countable sets. Assuming the axiom of countable choice, then a set is hereditarily countable...

Every second-countable space is hereditarily Lindelöf. Every countable space is hereditarily Lindelöf. Every Suslin space is hereditarily Lindelöf. Every...

in V [ G ] {\displaystyle V[G]} , not in V {\displaystyle V} . Hereditarily countable set Schindler, Ralf (2000), "Proper forcing and remarkable cardinals"...

countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets...

mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related...

Hereditarily P A space is hereditarily P for some property P if every subspace is also P. Hereditary A property of spaces is said to be hereditary if...

{\displaystyle \mathbb {N} } are called countable sets; these are either finite sets or countably infinite sets (sets of the same cardinality as N {\displaystyle...

intuitiveness. The language's alphabet consists of: A countably infinite amount of variables used for representing sets The logical connectives ¬ {\displaystyle \lnot...

uncountable ordinal is the set of all countable ordinals, expressed as ω1 or ⁠ Ω {\displaystyle \Omega } ⁠. In a well-ordered set, every non-empty subset...

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any...

Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus...

finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite"...

In mathematics, fuzzy sets (also known as uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently...

algebra of sets, completed to include countably infinite operations. Axiomatic set theory Image (mathematics) § Properties Field of sets List of set identities...

In set theory, a code for a hereditarily countable set x ∈ H ℵ 1 {\displaystyle x\in H_{\aleph _{1}}\,} is a set E ⊂ ω × ω {\displaystyle E\subset \omega...

A_{2}\cap A_{3}\cap \cdots } ". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras...

In set theory, the complement of a set A, often denoted by A ∁ {\displaystyle A^{\complement }} (or A′), is the set of elements not in A. When all elements...

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple...

mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure...

{\displaystyle {\mathsf {ZF}}} , this is the set H ℵ 1 {\displaystyle H_{\aleph _{1}}} of hereditarily countable sets and has ordinal rank at most ω 2 {\displaystyle...

between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships...