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#Q: How do you know that every term in the untyped lambda calculus has a fixed point?#

A: That's easy: let `T` be an arbitrary term in the lambda calculus.  If
`T` has a fixed point, then there exists some `X` such that `X <~~>
TX` (that's what it means to *have* a fixed point).

<pre><code>let L = \x. T (x x) in
let X = L L in
X &equiv; L L &equiv; (\x. T (x x)) L ~~> T (L L) &equiv; T X
</code></pre>

Please slow down and make sure that you understand what justified each
of the equalities in the last line.

#Q: How do you know that for any term `T`, `Y T` is a fixed point of `T`?#

A: Note that in the proof given in the previous answer, we chose `T`
and then set `X = L L = (\x. T (x x)) (\x. T (x x))`.  If we abstract over
`T`, we get the Y combinator, `\T. (\x. T (x x)) (\x. T (x x))`.  No matter
what argument `T` we feed `Y`, it returns some `X` that is a fixed point
of `T`, by the reasoning in the previous answer.

#Q: So if every term has a fixed point, even `Y` has fixed point.#

A: Right:

<pre><code>let Y = \T. (\x. T (x x)) (\x. T (x x)) in
Y Y &equiv; \T. (\x. T (x x)) (\x. T (x x)) Y
~~> (\x. Y (x x)) (\x. Y (x x))
~~> Y ((\x. Y (x x)) (\x. Y (x x)))
~~> Y (Y ((\x. Y (x x)) (\x. Y (x x))))
~~> Y (Y (Y (...(Y (Y Y))...)))</code></pre>


#Q: Ouch!  Stop hurting my brain.#

A: Is that a question?

Let's come at it from the direction of arithmetic.  Recall that we
claimed that even `succ`---the function that added one to any
number---had a fixed point.  How could there be an X such that X = X+1?
That would imply that

    X <~~> succ X <~~> succ (succ X) <~~> succ (succ (succ (X))) <~~> succ (... (succ X)...)

In other words, the fixed point of `succ` is a term that is its own
successor.  Let's just check that `X = succ X`:

<pre><code>let succ = \n s z. s (n s z) in
let X = (\x. succ (x x)) (\x. succ (x x)) in
succ X 
&equiv; succ ( (\x. succ (x x)) (\x. succ (x x)) ) 
~~> succ (succ ( (\x. succ (x x)) (\x. succ (x x))))
&equiv; succ (succ X)
</code></pre>

You should see the close similarity with `Y Y` here.