Axiom of adjunction explained
In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ given by "adjoining" the set y to the set x. It is stated as
\forallx.\forally.\existsw.\forallz.(z\inw\leftrightarrow(z\inx\lorz=y)).
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.
Interpretability of arithmetic
Tarski and Szmielew 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 .In fact, empty set and adjunction alone (without extensionality) suffice to interpret
}.
[1] (They are mutually interpretable.)
Adding epsilon-induction to empty set and adjunction yields a theory that is mutually interpretable with Peano arithmetic (
}).Another
axiom schema also yields a theory that is mutually interpretable with
}:
[2] \forallx.\forally.\existsw.\forallz.(z\inw\leftrightarrow((z\inx\lorz=y)\land\phi))
, where
is not allowed to have
free. This combines axioms of set theory: For
trivially true it reduced to the adjunction axiom above, and for
it gives the axiom of separation with
.
References
- Book: 978-0-8218-1041-5 . A Formalization of Set Theory without Variables . Tarski . Alfred . Givant . Steven R. . 1987 . American Mathematical Soc.. AMS Colloquium Publications, v. 41 .
Notes and References
- Mancini. Antonella. Montagna. Franco. Spring 1994. A minimal predicative set theory. Notre Dame Journal of Formal Logic. 35. 2. 186–203. 10.1305/ndjfl/1094061860. 23 November 2021. free.
- Web site: Issues in the foundations of mathematics. Friedman. Harvey M.. Harvey M. Friedman. June 2, 2002. Department of Mathematics. Ohio State University. January 18, 2023.
