AbbreviatedZF

AbbreviatedZF is a term used in some logic and set theory texts to refer to a compact, notationally streamlined presentation of Zermelo-Fraenkel set theory (ZF). It is not a distinct formal system; rather, it signals that the standard ZF axioms are being displayed in a condensed form. The idea is to reduce the burden of writing out every axiom schema while preserving the same mathematical content.

In practice, abbreviatedZF relies on the same language of first-order set theory with the membership relation and uses the core axioms of ZF—extensionality, foundation, pairing, union, power set, and infinity—together with the axiom schemas of Separation and Replacement. The distinguishing feature is that these schemas are presented in abbreviated or schematic form rather than enumerated as an infinite list of individual instances. For example, a typical abbreviated statement may read as a single sentence indicating that for every formula φ with possibly parameter symbols, and for every set A, there exists a subset of A consisting of those elements x in A that φ(x) holds.

Usage and interpretation: abbreviatedZF is primarily a pedagogical and expository device designed to improve readability and reduce redundancy when discussing ZF. It is not universally standardized; authors differ in how aggressively they summarize schemas or in what extra shorthand they employ. When used carefully, abbreviatedZF conveys the same theorems and proof obligations as standard ZF, since the underlying axioms are the same, just presented more compactly.

See also Zermelo-Fraenkel set theory, axiom schema, formalization, set theory.