Transitive model: Difference between revisions

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In matheamticalmathematical [[set theory]], a '''transitive model''' is a [[Model theory|model]] of set theory that is [[Inner model|standard and transitive]]. Standard means that the membership relation is the usual one, and transitive means that the model is a [[transitive set]] or class.

==ReferencesExamples==

*An [[inner model]] is a transitive model containing all ordinals.

*A countable transitive model (CTM) is, as the name suggests, a transitive model with a countable number of elements.

==Properties==

If ''M'' is a transitive model, then ω^''M'' is the standard ω. This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts. Each real number in a transitive model is a standard real number, although not all standard reals need be included in a particular transitive model.

==References==

* {{cite book | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition=Third Millennium | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 }}

[[Category:Set theory]]

[[Category:model theory]]