Transitive set explained

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 .Similarly, a class M is transitive if every element of M is a subset of M .

Examples

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

V _{\alpha} and L _{\alpha} leading to the construction of the von Neumann universe V and Gödel's constructible universe L are transitive sets. The universes V and L themselves are transitive classes.

This is a complete list of all finite transitive sets with up to 20 brackets:^[1]

\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}, \{\{\{\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\{\}\}, \{\{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\{\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}, \{\{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\{\{\}\}\}, \{\{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\{\{\}\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}, \{\{\}, \{\{\}, \{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\{\}, \{\{\}\}\}\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\{\}\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}, \{\{\{\}\}, \{\{\}, \{\{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\}\}, \{\{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\{\}, \{\{\}\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\{\}\}, \{\{\}, \{\{\}\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\{\{\{\}\}\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}\}\}, \{\{\}, \{\{\}\}, \{\{\{\}\}\}, \{\{\{\{\}\}\}\}, \{\{\}, \{\{\}\}\}, \{\{\}, \{\{\{\}\}\}\}\}.

Properties

A set

X is transitive if and only if $\backslash bigcup\; X\; \backslash subseteq\; X$, where $\backslash bigcup\; X$ is the union of all elements of X that are sets, $\backslash bigcup\; X\; =\; \backslash$.

If

X is transitive, then $\backslash bigcup\; X$ is transitive.

If

X and Y are transitive, then X\cup Y and X \cup Y \cup \{X,Y\} are transitive. In general, if Z is a class all of whose elements are transitive sets, then $\backslash bigcup\; Z$ and $Z\backslash cup\backslash bigcup\; Z$ are transitive. (The first sentence in this paragraph is the case of Z = \{X,Y\} .)

A set

X that does not contain urelements is transitive if and only if it is a subset of its own power set, $X\; \backslash subseteq\; \backslash mathcal(X).$ The power set of a transitive set without urelements is transitive.

Transitive closure

The transitive closure of a set

X is the smallest (with respect to inclusion) transitive set that includes X (i.e. $X\; \backslash subseteq\; \backslash operatorname(X)$).^[2] Suppose one is given a set X , then the transitive closure of X is

$$\backslash operatorname(X)\; =\; \backslash bigcup\; \backslash left\backslash .$$

Proof. Denote $X\_0\; =\; X$ and $X\_\; =\; \backslash bigcup\; X\_n$. Then we claim that the set

$$T\; =\; \backslash operatorname(X)\; =\; \backslash bigcup\_^\backslash infty\; X\_n$$

is transitive, and whenever $T\_1$ is a transitive set including $X$ then $T\; \backslash subseteq\; T\_1$.

Assume $y\; \backslash in\; x\; \backslash in\; T$. Then $x\; \backslash in\; X\_n$ for some $n$ and so $y\; \backslash in\; \backslash bigcup\; X\_n\; =\; X\_$. Since $X\_\; \backslash subseteq\; T$, $y\; \backslash in\; T$. Thus $T$ is transitive.

Now let $T\_1$ be as above. We prove by induction that $X\_n\; \backslash subseteq\; T\_1$ for all

n , thus proving that $T\; \backslash subseteq\; T\_1$: The base case holds since $X\_0\; =\; X\; \backslash subseteq\; T\_1$. Now assume $X\_n\; \backslash subseteq\; T\_1$. Then $X\_\; =\; \backslash bigcup\; X\_n\; \backslash subseteq\; \backslash bigcup\; T\_1$. But $T\_1$ is transitive so $\backslash bigcup\; T\_1\; \backslash subseteq\; T\_1$, hence $X\_\; \backslash subseteq\; T\_1$. This completes the proof.

Note that this is the set of all of the objects related to

X 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.

The transitive closure of a set can be expressed by a first-order formula:

x is a transitive closure of y iff x is an intersection of all transitive supersets of y (that is, every transitive superset of y contains x as a subset).

Transitive models of set theory

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

l{C} is defined to be strongly transitive if, for each set S\in l{C} , there exists a transitive superset T with S\subseteq T\subseteq l{C} . A strongly transitive class is automatically transitive. This strengthened transitivity assumption allows one to conclude, for instance, that l{C} contains the domain of every binary relation in l{C} .^[3]

See also

• End extension
• Transitive relation
• Supertransitive class

Notes and References

1. Web site: Number of rooted identity trees with n nodes (rooted trees whose automorphism group is the identity group).. OEIS.
2. Book: Ciesielski, Krzysztof . Set theory for the working mathematician . 1997 . Cambridge University Press . 978-1-139-17313-1 . Cambridge . 164 . 817922080.
3. Goldblatt (1998) p.161