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.

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

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 (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 Q.^[1] (They are mutually interpretable.)

Adding epsilon-induction to empty set and adjunction yields a theory that is mutually interpretable with Peano arithmetic (PA).

Combining adjunction and separation in the form of the single axiom schema ${\displaystyle \forall x\forall y\exists z\forall w(w\in z\leftrightarrow ((z\in x\lor z=y)\land \phi ))}$, where ${\displaystyle \phi }$ does not have ${\displaystyle z}$ free, also yields a theory that is mutually interpretable with PA.^[2]

References

• 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.

This set theory-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e

This page was last edited on 29 January 2023, at 07:08