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diff --git a/topics/week2_lambda_intro.mdwn b/topics/week2_lambda_intro.mdwn
index 5fd3a837..6bbb5399 100644
--- a/topics/week2_lambda_intro.mdwn
+++ b/topics/week2_lambda_intro.mdwn
@@ -1,13 +1,12 @@
-## Syntax of Lambda Calculus ##
+# Introduction to the Lambda Calculus #
 
 We often talk about "*the* Lambda Calculus", as if there were just
 one; but in fact, there are many, many variations.  The one we will
 start with, and will explore in some detail, is often called "the pure"
-or "the untyped" Lambda Calculus. Actually, there are many variations even under
-that heading. But all of the variations share a strong family
+or "the untyped" Lambda Calculus. Actually, there are many variations even under that heading. But all of the variations share a strong family
 resemblance, so what we learn now will apply to all of them.
 
-> Fussy note: calling this/these the "pure" lambda calculus is entrenched terminology,
+> Fussy note: calling this the "pure" Lambda Calculus is entrenched terminology,
 but it coheres imperfectly with other uses of "pure" we'll encounter. There are
 three respects in which the lambda calculus we'll be presenting might claim to
 deserve the name "pure": (1) it has no pre-defined constants and a very spare
@@ -15,16 +14,12 @@ syntax; (2) it has no types; (3) it has no side-effects, and is insensitive to
 the order of evaluation.
 
 > Sense (3) corresponds most closely to the other uses of "pure" you'll
-see in the surrounding literature. With respect to this point, it may be true that
-this lambda calculus has no side effects. (Let's revisit that assumption
-at the end of term.) But as we'll see next week, it is *not* true that it's insensitive
-to the order of evaluation. So if that's what we mean by "pure", this lambda
-calculus isn't as pure as you might hope to get. Some *typed* lambda calculi will
-turn out to be more pure in that respect.
+see in the surrounding literature. With respect to this point, it may be true that the Lambda Calculus has no side effects. (Let's revisit that assumption
+at the end of term.) But as we'll see next week, it is *not* true that it's insensitive to the order of evaluation. So if that's what we mean by "pure", this lambda calculus isn't as pure as you might hope to get. Some *typed* lambda calculi will turn out to be more pure in that respect.
 
 > But this lambda calculus is at least "pure" in sense (2). At least, it
-doesn't *explicitly talk about* any types. Some prefer to say that this
-lambda calculus *does* have types implicitly, it's just that
+doesn't *explicitly talk about* any types. Some prefer to say that the
+Lambda Calculus *does* have types implicitly, it's just that
 there's only one type, so that every expression is a member of
 that one type. If you say that, you have to say that functions from
 this type to this type also belong to this type. Which is weird... In
@@ -41,13 +36,15 @@ as you might hope to get.
 or "the untyped" Lambda Calculus, or even just "the" Lambda Calculus, this
 is the system that people will understand you to be referring to.
 
+## Syntax ##
+
 Here is its syntax:
 
 <blockquote>
-<strong>Variables</strong>: <code>x</code>, <code>y</code>, <code>z</code>...
+<strong>Variables</strong>: <code>x</code>, <code>y</code>, <code>z</code> ...
 </blockquote>
 
-Each variable is an expression. For any expressions M and N and variable a, the following are also expressions:
+Each variable is an expression. For any variable `a` and (possibly complex) expressions `M` and `N`, the following are also expressions:
 
 <blockquote>
 <strong>Abstract</strong>: <code>(&lambda;a M)</code>
@@ -66,13 +63,13 @@ Expressions in the lambda calculus are called "terms".  Here is the
 syntax of the lambda calculus given in the form of a context-free
 grammar:
 
-    T --> Var
-    T --> ( lambda Var T)
-    T --> ( T T )
-    Var --> x
-    Var --> y
-    Var --> z
-    ...
+> T --> Var  
+> T --> ( &lambda; Var T)  
+> T --> ( T T )  
+> Var --> x  
+> Var --> y  
+> Var --> z  
+> ...
 
 Very, very simple.
 
@@ -107,11 +104,11 @@ called the **argument**, and `M` is called the **body**.
 
 The rule of beta-reduction permits a transition from that expression to the following:
 
-    M [a <-- N]
+> <code>M</code> [<code>a</code> <-- <code>N</code>]
 
 What this means is just `M`, with any *free occurrences* inside `M` of
 the variable `a` replaced with the term `N` (--- "without capture", which
-we'll explain in the [[advanced notes|FIXME]]).
+we'll explain in the [[advanced notes|week2_lambda_advanced]]).
 
 What is a free occurrence?
 
@@ -160,7 +157,7 @@ say "reduction" for one or more reduction steps. So when we write:
 
     M ~~> N
 
-we'll mean that you can get from M to N by one or more reduction
+we'll mean that you can get from `M` to `N` by one or more reduction
 steps.
 
 When `M` and `N` are such that there is some common term (perhaps just one
@@ -171,9 +168,7 @@ like this:
     M <~~> N
 
 More details about the notation and metatheory of
-the lambda calculus here:
-
-*      [[ week2 lambda calculus fine points]]
+the lambda calculus are in [[this week's advanced notes|topics/week2_lambda_advanced]].
 
 
 ## Shorthand ##
@@ -203,9 +198,7 @@ and:
     z (\x (x y))
 
 
-**Dot notation** Dot means "assume a left paren here, and the matching right
-paren as far the right as possible without creating unbalanced
-parentheses". So:
+**Dot notation** Dot means "Insert a left parenthesis here, and the matching right parenthesis as far to the right as possible without creating unbalanced parentheses---so long as doing so would enclose an application or abstract not already wrapped in parentheses." Thus:
 
     \x (\y (x y))
 
@@ -233,7 +226,15 @@ abbreviates:
 
     ((\x (\y (x y))) x)
 
-We didn't have to insert any parentheses around the inner body of `\y. (x y)` because they were already there.
+The outermost parentheses were added because we have an application. `(\x. \y.
+...)` became `(\x (\y. ...))` because of the rule for dots. We didn't
+insert any parentheses around the inner body of `\y. (x y)` because they were
+already there. That is, in expressions of the form `\y. (...)`, the dot abbreviates
+nothing. It's harmless to write such a dot, though, and it can be conceptually
+helpful especially in light of the next convention.
+
+Similarly, we permit `\x. x`, which is shorthand for `\x x`, not for `\x (x)`, which
+our syntax forbids. (The [[lambda evaluator|/code/lambda_evaluator]] however tolerates such expressions.)
 
 
 **Merging lambdas** An expression of the form `(\x (\y M))`, or equivalently, `(\x. \y. M)`, can be abbreviated as:
@@ -252,7 +253,7 @@ function?  One popular answer is that a function can be represented by
 a set of ordered pairs.  This set is called the **graph** of the
 function.  If the ordered pair `(a, b)` is a member of the graph of `f`,
 that means that `f` maps the argument `a` to the value `b`.  In
-symbols, `f: a` &mapsto; `b`, or `f (a) == b`.  
+symbols, `f: a` &mapsto; `b`, or `f (a) == b`.
 
 In order to count as a *function* (rather
 than as merely a more general *relation*), we require that the graph not contain two
@@ -262,10 +263,10 @@ a unique result.
 
 The lambda calculus seems to be wonderfully well-suited for
 representing functions.  In fact, the untyped
-lambda calculus is Turing Complete (see [[!wikipedia Turing Completeness]]):
+lambda calculus is Turing Complete (see [[!wikipedia Turing completeness]]):
 all (recursively computable) functions can be represented by lambda
 terms. Which, by most people's lights, means that all functions we can "effectively decide" ---
-that is, compute in a mechanical way without requiring any ingenuity or insight ---
+that is, always apply in a mechanical way without requiring any ingenuity or insight, and be guaranteed of a correct answer after some finite number of steps ---
 can be represented by lambda terms. As we'll see, though, it will be fun
 (that is, not straightforward) unpacking how these things can be so "represented."
 
@@ -275,7 +276,7 @@ For some lambda terms, it is easy to see what function they represent:
 simply returns `M`: `((\x x) M) ~~> M`.
 
 > `(\x (x x))` duplicates its argument (applies it to itself):
-`((\x (x x)) M) ~~> (M M)` <!-- **M** or &omega;; W is \uv.uvv -->
+`((\x (x x)) M) ~~> (M M)` <!-- **M** or &omega;; W is \uv.uvv, L is \uv.u(vv) -->
 
 > `(\x (\y (y x)))` reorders its two arguments:
 `(((\x  (\y (y x))) M) N) ~~> (N M)` <!-- **T**; C is \uvx.uxv -->
@@ -288,7 +289,7 @@ lambda calculus, note that duplicating, reordering, and deleting
 elements is all that it takes to simulate the behavior of a general
 word processing program.  That means that if we had a big enough
 lambda term, it could take a representation of *Emma* as input and
-produce *Hamlet* as a result. 
+produce *Hamlet* as a result.
 
 Some of these functions are so useful that we'll give them special
 names.  In particular, we'll call the identity function `(\x x)`
@@ -306,7 +307,7 @@ and:
     (\z z)
 
 both represent the same function, the identity function. However, we said
-(FIXME in the advanced notes) that we would be regarding these expressions as
+[[in the advanced notes|week2_lambda_advanced]] that we would be regarding these expressions as
 synactically equivalent, so they aren't yet really examples of *distinct*
 lambda expressions representing a single function. However, all three of these
 are distinct lambda expressions:
@@ -317,7 +318,7 @@ are distinct lambda expressions:
 
     (\z z)
 
-yet when applied to any argument M, all of these will always return M. So they
+yet when applied to any argument `M`, all of these will always return `M`. So they
 have the same extension. It's also true, though you may not yet be in a
 position to see, that no other function can differentiate between them when
 they're supplied as an argument to it. However, these expressions are all
@@ -346,8 +347,7 @@ embodied in the lambda calculus.
 mathematical conception permits many *uncomputable* functions, but the
 lambda calculus can't express those.
 
-2.  More problematically, some lambda terms express "functions" that can take
-themselves as arguments.  If we wanted to represent that set-theoretically, and
+2.  More problematically, lambda terms express "functions" that can *take themselves* as arguments.  If we wanted to represent that set-theoretically, and
 identified functions with their extensions, then we'd have to have some
 extension that contained (an ordered pair containing) itself as a member. Which
 we're not allowed to do in mainstream set-theory.  But in the lambda calculus
@@ -357,8 +357,9 @@ this is permitted and common --- and in fact will turn out to be indispensable.
 
         (\x x) (\x x)
 
-   This is a redex that reduces to the identity function (of course). We can
-apply the **K** function to another argument and itself:
+   This is a redex that reduces to the identity function (of course).
+
+   We can apply the **K** function to another argument and itself:
 
    > <code><b>K</b> z <b>K</b></code>
 
@@ -376,7 +377,7 @@ In fact it *does* turn out to be possible to represent the Lambda Calculus
 set-theoretically. But not in the straightforward way that identifies
 functions with their graphs. For years, it wasn't known whether it would be possible to do this. But then
 [[!wikipedia Dana Scott]] figured out how to do it in late 1969, that is,
-he formulated the first "denotational semantics" for this lambda calculus.
+he formulated the first "denotational semantics" for this Lambda Calculus.
 Scott himself had expected that this wouldn't be possible to do. He argued
 for its unlikelihood in a paper he wrote only a month before the discovery.
 
@@ -413,11 +414,15 @@ also count these functions:
 as equivalent. This additional move is called **eta-reduction**. It's
 crucial to eta-reduction that the outermost variable binding in the
 abstract we begin with (here, `\x`) be of a variable that occurs free
-exactly once in the body of that abstract, and that it be in the
-rightmost position.
+exactly once in the body of that abstract, and that that free occurrence be the rightmost outermost constituent.
 
-In the extended proof theory/theories we get be permitting eta-reduction/conversion
-as well as beta-reduction, *all computable functions with the same
+The expression:
+
+    (\x (\y (y x))
+
+can't be eta-reduced, because the rightmost outermost constituent is not `x` but `(\y (y x)`.
+
+In the extended proof theory/theories we get be permitting eta-reduction/conversion as well as beta-reduction, *all computable functions with the same
 extension do turn out to be equivalent*, that is, convertible.
 
 However, we still shouldn't assume we're working with functions
@@ -428,18 +433,20 @@ reasons sketched above.
 ## The analogy with `let` ##
 
 In our basic functional programming language, we used `let`
-expressions to assign values to variables.  For instance, 
+expressions to assign values to variables.  For instance,
 
     let x match 2
-    in (x, x) 
+    in (x, x)
 
-evaluates to the ordered pair (2, 2).  It may be helpful to think of
+evaluates to the ordered pair `(2, 2)`.  It may be helpful to think of
 a redex in the lambda calculus as a particular sort of `let`
 construction.
 
-    ((\x BODY) ARG) is analogous to
+    ((\x BODY) ARG)
+
+is analogous to
 
-    let x match ARG 
+    let x match ARG
     in BODY
 
 This analogy should be treated with caution.  For one thing, our `letrec`
@@ -461,7 +468,7 @@ calculus, an abstract such as `(\x (x x))` is perfectly well-formed
 and coherent, but it is not possible to write a `let` expression that
 does not have an `ARG`. That would be like:
 
-&nbsp;&nbsp;&nbsp;&nbsp;`let x match` *missing*
+&nbsp;&nbsp;&nbsp;&nbsp;`let x match` *missing*  
 &nbsp;&nbsp;&nbsp;&nbsp;`in x x`
 
 Nevertheless, the correspondence is close enough that it can guide our