Cleanup

Advanced/Lambda Calculus/parts/04 recursion.tex

@@ -8,12 +8,13 @@ $$
 	\end{cases}
 $$
 
-We cannot re-create this in lambda notation. Functions in lambda calculus are \textit{anonymous}, which means we can't call them before they've been fully defined.
+We cannot re-create this in lambda calculus, since we aren't given a way to recursively call functions.
 
-\vspace{1ex}
+\vspace{2mm}
 
-As an example, consider the statement $A = \lm a. A~a$ \par
-This means \say{write $(\lm a.A~a)$ whenever you see $A$.} However, $A$ is \textit{inside} what we're rewriting. We'd fall into infinite recursion before even starting our $\beta$-reduction!
+One could think that $A = \lm a. A~a$ is a recursive function. In fact, it is not. \par
+Remember that such \say{definitions} aren't formal structures in lambda calculus. \par
+They're just shorthand that simplifies notation.
 
 \begin{instructornote}
 	We're talking about recursion, and \textit{computability} isn't far away. At one point or another, it may be good to give the class a precise definition of \say{computable by lambda calculus:}
@@ -29,12 +30,12 @@ This means \say{write $(\lm a.A~a)$ whenever you see $A$.} However, $A$ is \text
 
 \problem{}
 Write an expression that resolves to itself. \par
-\note{Your answer should be short and sweet.}
+\note{Your answer should be quite short.}
 
 \vspace{1ex}
 
 This expression is often called $\Omega$, after the last letter of the Greek alphabet. \par
-$\Omega$ useless on its own, but gives us a starting point for recursion.
+$\Omega$ useless on its own, but it gives us a starting point for recursion.
 
 \begin{solution}
 	$\Omega = M~M = (\lm x . xx) (\lm x . xx)$