Axiom of adjunction

Tarski and Szmielew showed that Robinson arithmetic (${\displaystyle {\mathsf {Q}}}$) can be interpreted in a weak set theory whose axioms are extensionality, the existence of the empty set, and the axiom of adjunction (Tarski 1953, p.34). In fact, empty set and adjunction alone (without extensionality) suffice to interpret ${\displaystyle {\mathsf {Q}}}$.[1] (They are mutually interpretable.)

Adding epsilon-induction to empty set and adjunction yields a theory that is mutually interpretable with Peano arithmetic (${\displaystyle {\mathsf {PA}}}$). Another axiom schema also yields a theory that is mutually interpretable with ${\displaystyle {\mathsf {PA}}}$:[2]

${\displaystyle \forall x.\forall y.\exists w.\forall z.{\Big (}z\in w\leftrightarrow {\big (}(z\in x\lor z=y)\land \phi {\big )}{\Big )}}$,

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

• Bernays, Paul (1937), "A System of Axiomatic Set Theory--Part I", The Journal of Symbolic Logic, Association for Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862
• Kirby, Laurence (2009), "Finitary Set Theory", Notre Dame J. Formal Logic, 50 (3): 227–244, doi:10.1215/00294527-2009-009, MR 2572972
• Tarski, Alfred (1953), Undecidable theories, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland Publishing Company, MR 0058532
• Tarski, A., and Givant, Steven (1987) A Formalization of Set Theory without Variables. Providence RI: AMS Colloquium Publications, v. 41.