In mathematical set theory, theaxiom of adjunction states that for any two setsx,y there is a setw = x ∪ {y} given by "adjoining" the sety to the setx. It is stated as

${\displaystyle \mathrm{\forall }x.\mathrm{\forall }y.\mathrm{\exists }w.\mathrm{\forall }z.(z\in w\leftrightarrow (z\in x\vee z=y)).}$

Bernays (1937, page 68, axiom II (2)) introduced the axiom of adjunction as one of the axioms for a system of set theory that he introduced in about 1929.It is a weak axiom, used in some weak systems of set theory such asgeneral set theory orfinitary set theory. The adjunction operation is also used as one of the operations ofprimitive recursive set functions.

Tarski andSzmielew showed thatRobinson arithmetic (${\displaystyle \mathsf{Q}}$) can be interpreted in a weak set theory whose axioms areextensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34).In fact, in the language of set theory extended with a function symbol for adjunction, the axioms of empty set and adjunction alone (without extensionality) suffice to interpret${\displaystyle \mathsf{Q}}$.^[1] (They are mutually interpretable.)

Addingepsilon-induction to empty set and adjunction yields a theory that is mutually interpretable withPeano arithmetic (${\displaystyle \mathsf{P}\mathsf{A}}$).Anotheraxiom schema also yields a theory that is mutually interpretable with${\displaystyle \mathsf{P}\mathsf{A}}$:^[2]

${\displaystyle \mathrm{\forall }x.\mathrm{\forall }y.\mathrm{\exists }w.\mathrm{\forall }z.(z\in w\leftrightarrow ((z\in x\vee z=y)\wedge \varphi ))}$,

where${\displaystyle \varphi }$ is not allowed to have${\displaystyle w}$ free. This combines axioms of set theory: For${\displaystyle \varphi }$ trivially true it reduced to the adjunction axiom above, and for${\displaystyle (z\ne y)\wedge P}$ it gives theaxiom of separation with${\displaystyle P}$.

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