End extension
Extension of a transitive set From Wikipedia, the free 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
- is a substructure of , (i.e., and ), and
- whenever and hold, i.e., no new elements are added by to the elements of .[1]
The second condition can be equivalently written as for all .
For example, is an end extension of if and are transitive sets, and .
A related concept is that of a top extension (also known as rank extension), where a model is a top extension of a model if and for all and , we have , where 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.[2] If the elementarity requirement is weakened to being elementary for formulae that are on the Lévy hierarchy, every countable structure in which -collection holds has a -elementary end extension.[3]
References
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