End extension
Encyclopedia
In model theory
and set theory
, which are disciplines within mathematics, a model
of some axiom system of set theory
in the language of set theory is an end extension of 
, in symbols 
, if
The following is an equivalent definition of end extension:
is a substructure
of
, and 
for all 
.
For example,
is an end extension of 
if 
and 
are transitive set
s, and
.
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
and 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...
, which are disciplines within mathematics, a model
of some axiom system of set theorySet 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...
in the language of set theory is an end extension of , in symbols , if
-
is a substructureSubstructureIn mathematical logic, an substructure or subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure...
of
, and -
whenever 
and 
hold, i.e., no new elements are added by 
to the elements of 
.
The following is an equivalent definition of end extension:
is a substructureSubstructure
In mathematical logic, an substructure or subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure...
of
, and for all .For example,
is an end extension of if and are transitive setTransitive set
In set theory, a set A is transitive, if* whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,* whenever x ∈ A, and x is not an urelement, then x is a subset of A....
s, and
.