Axiom of adjunction

In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x.

$\forall x\,\forall y\,\exists w\,\forall z\,[z\in w\leftrightarrow (z\in x\lor z=y)].$

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 as general set theory or finitary set theory. The adjunction operation is also used as one of the operations of primitive recursive set functions.

Tarski and Smielew showed that Robinson arithmetic 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 Robinson arithmetic.

References

• Bernays, Paul (1937), "A System of Axiomatic Set Theory--Part I", The Journal of Symbolic Logic, 2 (1), Association for Symbolic Logic: 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.