Y = IMG:lambda-6.gif:_f_.(IMG:lambda-6.gif:_x_.(_f_ (_x_ _x_)) IMG:lambda-6.gif:_x_.(_f_ (_x_ _x_)))

 Y = f.(x.(f (x x)) x.(f (x x)))

• Actually many others are possible, but this is the simplest

• Claim: (Y f) = (f (Y f))

• Let's see:

 (Y f)

 (f.(x.(f (x x)) x.(f (x x))) f)

 (x.(f (x x)) x.(f (x x)))

 (f (x.(f (x x)) x.(f (x x)))) 

• Why look, that's exactly (f (Y f)).

• This is f making a recursive call

• If another recursive call is needed the new (Y f) will generate one

• Etc.