End extension: Difference between revisions
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==Existence== |
==Existence== |
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Keisler and Morley showed that every countable model of ZF has an end extension of which it is an [[elementary substructure]].<ref> |
Keisler and Morley showed that every countable model of ZF has an end extension of which it is an [[elementary substructure]].<ref>{{citation |
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| last1=Keisler | first1=H. Jerome |
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| last2=Morley | first2=Michael |
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| title=Elementary extensions of models of set theory |
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| journal=[[Israel Journal of Mathematics]] |
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| volume=5 |
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| date=1968 |
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| pages=49–65 |
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| doi=10.1007/BF02771605}}</ref> If the elementarity requirement is weakened to being elementary for formulae that are <math>\Sigma_n</math> on the [[Lévy hierarchy]], every countable structure in which <math>\Sigma_n</math>-collection holds has a <math>\Sigma_n</math>-elementary end extension.<ref>{{citation |
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| last1=Kaufmann | first1=Matt |
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| contribution=On existence of Σ<sub>n</sub> end extensions |
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| series=Lecture Notes in Mathematics |
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| title=Logic Year 1979–80 |
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| date=1981 |
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| pages=92–103 |
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| isbn=3-540-10708-8 |
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| volume=859 |
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| doi=10.1007/BFb0090942}}</ref> |
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==References== |
==References== |
Revision as of 16:58, 28 August 2023
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 .
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.[1] 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.[2]
References
- ^ Keisler, H. Jerome; Morley, Michael (1968), "Elementary extensions of models of set theory", Israel Journal of Mathematics, 5: 49–65, doi:10.1007/BF02771605
- ^ 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