Search Results for “diagonal intersection”

Source: Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.
If

δ

{\displaystyle \displaystyle \delta }

is an ordinal number and

⟨

X

α

∣
α
<
δ
⟩

{\displaystyle \displaystyle \langle X_{\alpha }\mid \alpha <\delta \rangle }

is a sequence of subsets of

δ

{\displaystyle \displaystyle \delta }

, then the diagonal intersection, denoted by

Δ

α
<
δ

X

α

,

{\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha },}

is defined to be

{
β
<
δ
∣
β
∈

⋂

α
<
β

X

α

}
.

{\displaystyle \displaystyle \{\beta <\delta \mid \beta \in \bigcap _{\alpha <\beta }X_{\alpha }\}.}

That is, an ordinal

β

{\displaystyle \displaystyle \beta }

is in the diagonal intersection

Δ

α
<
δ

X

α

{\displaystyle \displaystyle \Delta _{\alpha <\delta }X_{\alpha }}

if and only if it is contained in the first

β

{\displaystyle \displaystyle \beta }

members of the sequence. This is the same as

⋂

α
<
δ

(
[
0
,
α
]
∪

X

α

)
,

{\displaystyle \displaystyle \bigcap _{\alpha <\delta }([0,\alpha ]\cup X_{\alpha }),}

where the closed interval from 0 to

α

{\displaystyle \displaystyle \alpha }

is used to
avoid restricting the range of the intersection.