Transitive set

Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class.

Any of the stages and leading to the construction of the von Neumann universe and Gödel's constructible universe are transitive sets. The universes and themselves are transitive classes.

This is a complete list of all finite transitive sets with up to 20 pairs of brackets:

A set is transitive if and only if , where is the union of all elements of that are sets; formally,

.

Moreover, if is transitive, then is transitive.

If and are transitive, then and are transitive. In general, if is a class all of whose elements are transitive sets, then and are transitive. (The first sentence in this paragraph is the case of .)

A set that does not contain urelements is transitive if and only if it is a subset of its own power set, . The power set of a transitive set without urelements is transitive.

The transitive closure of a set , denoted , is the smallest (with respect to inclusion) transitive set that includes . The transitive closure of can be equivalently defined as

Proof. Denote and . Then we claim that the set

is transitive, and whenever is a transitive set including then .

Assume . Then for some and so . Since , . Thus is transitive.

Now let be as above. We prove by induction that for all , thus proving that . The base case holds since . Now assume . Then . But is transitive so , hence . This completes the proof.

Note that this is the set of all of the objects related to by the transitive closure of the membership relation, since the union of a set can be expressed in terms of the relative product of the membership relation with itself.

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models. The reason is that properties defined by bounded formulas are absolute for transitive classes.

A transitive set (or class) that is a model of a formal system of set theory is called a transitive model of the system (provided that the element relation of the model is the restriction of the true element relation to the universe of the model). Transitivity is an important factor in determining the absoluteness of formulas.

In the superstructure approach to non-standard analysis, the non-standard universes satisfy strong transitivity. Here, a class is defined to be strongly transitive if, for each set , there exists a transitive superset with . A strongly transitive class is automatically transitive. This strengthened transitivity assumption allows one to conclude, for instance, that contains the domain of every binary relation in .