 Here \(w = \{ x \mid  P(x) \}\) becomes the set of all reals definable by phrases in English. For any denumerable subset \(y\) of \(w, \delta(y)\) is a real that by construction will differ from all reals in \(y\) (it differs from the \(n\)th real in \(y\) on the \(n\)th decimal place). Letting \(y\) equal \(w\) we thus get \(\delta(w) \not\in w\). However, at the same time \(\delta(w)\) is definable by a phrase in English, so \(\delta(w) \in w\), and we have a contradiction. This contradiction is Richard’s paradox. 