Diagonal intersection
Encyclopedia
Diagonal intersection is a term used in mathematics
, especially in set theory
.
If is an ordinal number
and
is a sequence
of subsets of , then the diagonal intersection, denoted by
is defined to be
That is, an ordinal
is in the diagonal intersection iff it is contained in the first members of the sequence. This is the same as
where the closed interval from 0 to is used to
avoid restricting the range of the intersection.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
.
If is an ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
and
is a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of subsets of , then the diagonal intersection, denoted by
is defined to be
That is, an ordinal
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
is in the diagonal intersection iff it is contained in the first members of the sequence. This is the same as
where the closed interval from 0 to is used to
avoid restricting the range of the intersection.