nLab transitive set

Definition

In unsorted material set theory, a pure set $S$ is called transitive if $a\in b\in S$ implies that $a\in S$. Note that this does not mean that $\in$ is a transitive binary relation on $S$ itself. (In fact, assuming the axiom of foundation, $\in$ is a transitive relation on $S$ precisely when $S$ is a von Neumann ordinal number.)

Transitive sets are a natural place to look for inner model?s of a material set theory: that is, sets $S$ such that $S$ together with the restriction of $\in$ to $S$ satisfies some or all of the axioms of the set theory. This is especially so because of Mostowski's collapsing lemma that any extensional well-founded relation is isomorphic to a transitive set.

Transitive closures

In ZFC, one can prove that every pure set $x$ is contained in a least transitive pure set, called its transitive closure. This can be defined as the set of all $y$ such that there is a chain $y = x_n \in \cdots \in x_0 = x$ for some natural number $n$. The proof that this set exists requires the axiom of replacement and the property of induction for unbounded formulas (which follows from the axiom of separation). In set theories that are too weak to prove the existence of transitive closures in this way, their existence is sometimes assumed explicitly, as the axiom of transitive closure.

Modeling pure sets

Given the existence of transitive closures, pure sets can be identified with subsets of transitive sets, and hence (given Mostowski’s lemma) with subsets of extensional well-founded relations. This latter characterization is completely structural, and thus can be used to model pure sets in structural set theory.

category: foundational axiom