ωω (\h \lst. (isempty lst) zero (add one ((h h) (extracttail lst))))

and this will indeed implement the recursive function we couldn't earlier figure out how to define.
In broad brushstrokes, `H` is half of the `get_length` function we're seeking, and `H` has the form:
@@ 416,7 +424,7 @@ to *the tail* of the list we were evaluating its application to at the previous
##Fixedpoint Combinators Are a Bit Intoxicating##
![tatoo](/ycombinator.jpg)
+![tatoo](/ycombinatorfixed.jpg)
There's a tendency for people to say "Ycombinator" to refer to fixedpoint combinators generally. We'll probably fall into that usage ourselves. Speaking correctly, though, the Ycombinator is only one of many fixedpoint combinators.
@@ 546,18 +554,17 @@ sentence in which it occurs, the sentence denotes a fixed point for
the identity function. Here's a fixed point for the identity
function:
 Y I
 (\f. (\h. f (h h)) (\h. f (h h))) I
 (\h. I (h h)) (\h. I (h h)))
 (\h. (h h)) (\h. (h h)))
 ω ω
 &Omega

+Y I
+(\f. (\h. f (h h)) (\h. f (h h))) I
+(\h. I (h h)) (\h. I (h h)))
+(\h. (h h)) (\h. (h h)))
+ω ω
+&Omega
+
Oh. Well! That feels right. The meaning of *This sentence is true*
in a context in which *this sentence* refers to the sentence in which
it occurs is Ω, our prototypical infinite loop...
+it occurs is Ω
, our prototypical infinite loop...
What about the liar paradox?
@@ 584,3 +591,50 @@ See Barwise and Etchemendy's 1987 OUP book, [The Liar: an essay on
truth and circularity](http://tinyurl.com/2db62bk) for an approach
that is similar, but expressed in terms of nonwellfounded sets
rather than recursive functions.
+
+##However...##
+
+You should be cautious about feeling too comfortable with
+these results. Thinking again of the truthteller paradox, yes,
+Ω
is *a* fixed point for `I`, and perhaps it has
+some a privileged status among all the fixed points for `I`, being the
+one delivered by Y and all (though it is not obvious why Y should have
+any special status).
+
+But one could ask: look, literally every formula is a fixed point for
+`I`, since
+
+ X <~~> I X
+
+for any choice of X whatsoever.
+
+So the Y combinator is only guaranteed to give us one fixed point out
+of infinitely manyand not always the intuitively most useful
+one. (For instance, the squaring function has zero as a fixed point,
+since 0 * 0 = 0, and 1 as a fixed point, since 1 * 1 = 1, but `Y
+(\x. mul x x)` doesn't give us 0 or 1.) So with respect to the
+truthteller paradox, why in the reasoning we've
+just gone through should we be reaching for just this fixed point at
+just this juncture?
+
+One obstacle to thinking this through is the fact that a sentence
+normally has only two truth values. We might consider instead a noun
+phrase such as
+
+(3) the entity that this noun phrase refers to
+
+The reference of (3) depends on the reference of the embedded noun
+phrase *this noun phrase*. It's easy to see that any object is a
+fixed point for this referential function: if this pen cap is the
+referent of *this noun phrase*, then it is the referent of (3), and so
+for any object.
+
+The chameleon nature of (3), by the way (a description that is equally
+good at describing any object), makes it particularly well suited as a
+gloss on pronouns such as *it*. In the system of
+[Jacobson 1999](http://www.springerlink.com/content/j706674r4w217jj5/),
+pronouns denote (you guessed it!) identity functions...
+
+Ultimately, in the context of this course, these paradoxes are more
+useful as a way of gaining leverage on the concepts of fixed points
+and recursion, rather than the other way around.