End extension explained

In model theory and set theory, which are disciplines within mathematics, a model

ak{B}=\langle B, F\rangle of some axiom system of set theory T in the language of set theory is an end extension of ak{A}=\langle A, E\rangle , in symbols ak{A}\subseteq _{endak{B}} , if ak{A} is a substructure of ak{B} , (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 ak{B} to the elements of A .^[1]

The second condition can be equivalently written as

\{b\in A : b E a\} = \{b\in B : b F a\} for all a\in A .

For example,

\langle B, \in\rangle is an end extension of \langle A, \in\rangle if A and B are transitive sets, and A\subseteq B .

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

ak{B}=\langle B, F\rangle is a top extension of a model ak{A}=\langle A, E\rangle if ak{A}\subseteq _{endak{B}} and for all a \in A and b \in B\setminus A , we have rank(b) > rank(a) , where rank( ⋅ ) denotes the rank of a set.

Existence

Keisler and Morley showed that every countable model of ZF has an end extension which is also an elementary extension. If the elementarity requirement is weakened to being elementary for formulae that are

\Sigma _{n} on the Lévy hierarchy, every countable structure in which \Sigma _{n} -collection holds has a \Sigma _{n} -elementary end extension.

Notes and References

1. H. J. Keisler, J. H. Silver, "End Extensions of Models of Set Theory", p.177. In Axiomatic Set Theory, Part 1 (1971), Proceedings of Symposia in Pure Mathematics, Dana Scott, editor.