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...

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, 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...

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...

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

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

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

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

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

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

|\mathbb {N} |=\aleph _{0}} are called countable sets; these are either finite sets or countably infinite sets (sets of cardinality ℵ 0 {\displaystyle \aleph...

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...

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...

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...

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...

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

Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included...

existence of ordinals α ≥ ω + ω, which include uncountably many hereditarily countable sets. This follows from Skolem's result that Vω+ω satisfies Zermelo's...

{\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...

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...

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...

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...

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

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...

exist sets which are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets...

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...

{Z} ,n=2k\}} The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation...

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be...