ZFCn

ZFCn is a term found in set-theory literature to denote either a finite fragment of ZFC or a family of theories ZFCn indexed by a natural number n. There is no universally standard definition, and the precise content of ZFCn varies by author. In practice, authors use ZFCn to study finite axiomatizations or bounded forms that approximate ZFC.

Common formulations fall into two broad patterns. One approach defines ZFCn as a finite fragment obtained by selecting a finite subset of the usual ZFC axioms, often together with standard axioms of infinity, pairing, union, power set, and so on, while treating the infinite axiom schemas (such as Replacement or Comprehension) in a restricted way. Another approach defines ZFCn by restricting the Replacement (and sometimes Separation) schemas to formulas of bounded complexity or to sets of bounded rank, yielding a finitely axiomatizable theory that approximates ZFC for certain ranges of sets.

Relation to full ZFC varies with the definition. For any fixed n, ZFCn is typically weaker than ZFC, since it uses a finite or bounded version of otherwise infinite schemas. As n increases (in definitions where n controls the bound), ZFCn can become closer in strength to ZFC, though it remains a separate theory for each finite n.

Use and motivation often center on proof-theoretic and foundational questions: examining how much of ZFC can be captured by a finite axiomatization, comparing relative strengths, and informing discussions about the necessity of the full Replacement or Comprehension schemas. Because the notation is not standardized, readers should consult the specific definition used in a given source.