Editing Rice's theorem (section)

==Proof by Kleene's recursion theorem==
Assume for contradiction that <math>P</math> is a non-trivial, extensional and computable set of natural numbers. There is a natural number <math>a \in P</math> and a natural number <math>b \notin P</math>. Define a function <math>Q</math> by <math>Q_e(x) = \varphi_b(x)</math> when <math>e \in P</math> and <math>Q_e(x) = \varphi_a(x)</math> when <math>e \notin P</math>. By [[Kleene's recursion theorem]], there exists <math>e</math> such that <math>\varphi_e = Q_e</math>. Then, if <math>e \in P</math>, we have <math>\varphi_e = \varphi_b</math>, contradicting the extensionality of <math>P</math> since <math>b \notin P</math>, and conversely, if <math>e \notin P</math>, we have <math>\varphi_e = \varphi_a</math>, which again contradicts extensionality since <math>a \in P</math>.