A similar phenomenon occurs in formalized theories that are able to refer to their own syntax, such as Zermelo–Fraenkel set theory (ZFC). Say that a formula φ("x") "defines a real number" if there is exactly one real number "r" such that φ("r") holds. Then it is not possible to define, by ZFC, the set of all (Gödel numbers of) formulas that define real numbers. For, if it were possible to define this set, it would be possible to diagonalize over it to produce a new definition of a real number, following the outline of Richard's paradox above. Note that the set of formulas that define real numbers may exist, as a set "F"; the limitation of ZFC is that there is not any formula that defines "F" without reference to other sets. This is related to Tarski's indefinability theorem.