Paarungsaxiom

The Paarungsaxiom, or Axiom of Pairing, is a fundamental axiom in Zermelo-Fraenkel set theory (ZF). It is one of the foundational principles that govern how sets can be constructed and manipulated within this axiomatic system. The axiom states that for any two sets, say A and B, there exists a unique set C whose elements are precisely A and B. This can be expressed formally as: for all sets $x$ and for all sets $y$, there exists a set $z$ such that for all sets $w$, $w \in z$ if and only if $w = x$ or $w = y$. This set $z$ is often denoted as $\{x, y\}$.

The Paarungsaxiom is crucial for building more complex sets from simpler ones. For example, it allows us to form the singleton set $\{x\}$ by pairing a set with itself, i.e., $\{x, x\}$. This is because the definition of set equality ensures that $\{x, x\}$ is the same set as $\{x\}$. The axiom also allows us to form the pair $\{x, y\}$ where $x$ and $y$ are distinct sets. Without the Paarungsaxiom, it would be impossible to guarantee the existence of such basic set constructions, which are essential for developing the rest of set theory. It is considered one of the simplest and most intuitive axioms of ZF, reflecting the basic idea that we can combine existing sets into new ones.