Transitive model

In mathematical set theory, a transitive model is a model of set theory that is 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.

Examples

• 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

• Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-44085-7. Zbl 1007.03002.