End Extension

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

and

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...

, which are disciplines within mathematics, a model

\mathfrak=\langle B, F\rangle

of some axiom system of

T

in the language of set theory is an end extension of

\mathfrak=\langle A, E\rangle

, in symbols

\mathfrak\subseteq_\text\mathfrak

, if #

\mathfrak

is a substructure of

\mathfrak

, (i.e.,

A \subseteq B

and

E = F, _A

), and #

b\in A

whenever

a\in A

and

bFa

hold, i.e., no new elements are added by

\mathfrak

to the elements of

A

. The second condition can be equivalently written as

\=\

for all

a\in A

. For example,

\langle B, \in\rangle

is an end extension of

\langle A, \in\rangle

A

and

B

are

transitive set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Si ...

s, and

A\subseteq B

. A related concept is that of a top extension (also known as rank extension), where a model

\mathfrak=\langle B, F\rangle

is a top extension of a model

\mathfrak=\langle A, E\rangle

\mathfrak\subseteq_\text\mathfrak

and for all

a \in A

and

b \in B\setminus A

, we have

rank(b) > rank(a)

, where

rank(\cdot)

denotes the

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

of a set. Mathematical logic Model theory Set theory {{mathlogic-stub