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

  1. is a substructure of , (i.e., and ), and
  2. 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 of which it is an elementary substructure.[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

  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.
  2. Keisler, H. Jerome; Morley, Michael (1968), "Elementary extensions of models of set theory", Israel Journal of Mathematics, 5: 49–65, doi:10.1007/BF02771605
  3. Kaufmann, Matt (1981), "On existence of Σn end extensions", Logic Year 1979–80, Lecture Notes in Mathematics, vol. 859, pp. 92–103, doi:10.1007/BFb0090942, ISBN 3-540-10708-8
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.