End extension: Difference between revisions
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⚫ | In [[model theory]] and [[set theory]], which are disciplines within mathematics, a model <math>\mathfrak{B}=\langle B, F\rangle </math> of some axiom system of [[set theory]] <math> T\, </math> in the language of set theory is an '''end extension''' of <math>\mathfrak{A}=\langle A, E\rangle </math>, in symbols <math>\mathfrak{A}\subseteq_\text{end}\mathfrak{B}</math>, if |
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* <math>\mathfrak{A}</math> is a [[substructure]] of <math>\mathfrak{B}</math>, and |
* <math>\mathfrak{A}</math> is a [[substructure]] of <math>\mathfrak{B}</math>, and |
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* <math> b\in A</math> whenever <math>a\in A</math> and <math>bFa\,</math> hold, i.e., no new elements are added by <math>\mathfrak{B}</math> to the elements of <math>\mathfrak{A}</math>. |
* <math> b\in A</math> whenever <math>a\in A</math> and <math>bFa\,</math> hold, i.e., no new elements are added by <math>\mathfrak{B}</math> to the elements of <math>\mathfrak{A}</math>. |
Revision as of 12:54, 28 January 2009
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 , and
- whenever and hold, i.e., no new elements are added by to the elements of .
The following is an equivalent definition of end extension: is a substructure of , and for all .
For example, is an end extension of if and are transitive sets, and .