Material set theory

Axiom Schema of Replacement

The Axiom of Replacement, technically an axiom schema, is a possible axiom of Material set theory suggested by Abraham Fraenkel in the early 1920’s¹: zf Let 𝜑(𝑥,𝑦) be a Class function, i.e. a predicative formula such that (∀𝑥)(∃!𝑦)𝜑(𝑥,𝑦). Then,

(∀𝔐⁡𝐴)(∃𝔐⁡𝐵)[𝑦∈𝐵⟺(∃𝑥∈𝐴)𝜑(𝑥,𝑦)]

which is to say, the image of a set under a mapping is a set.

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Footnotes

1. 2006. Notes on set theory, §11, p. 157 ↩