A fundamental concept in the theory of ordinal numbers is a transitive set, which we want now to introduce formally:
A transitive set \(Z\) is one in which the following implication is always fulfilled:
$$x\in y\wedge y\in Z\Longrightarrow x\in Z,$$
i.e. if $x$ is element of $y$ and $y$ is element of $Z$, then $x$ is also an element of $Z.$ This is equivalent to the following: If $y\in X$, then $y\subseteq X$ (i.e. every element $y$ of $Z$ is also its subset).
Because of the axiom of foundation, by which no set can contain itself, we can even require that $y$ is a proper subset of $Z,$ and the definition of a transitive set can then be written as follows: $$y\in Z\Longrightarrow y\subset Z.$$
Corollaries: 1
Corollaries: 1 2
Definitions: 3 4
Lemmas: 5
Motivations: 6 7
Proofs: 8 9 10 11 12 13 14 15 16
Propositions: 17
Theorems: 18