In mathematical
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, a transitive model is a
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
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
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold:
* whenever x \in A, and y \in x, then y \in A.
* whenever x \in A, and x is not an urelement, then x is a subset of A.
Si ...
or class.
Examples
*An
inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle b ...
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
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-44085-7 , year=2003 , zbl=1007.03002
Set theory