Is the Y Combinator 3rd order / in lambda calculus ?
Клод 3.7 =>
Yes, the Y Combinator is indeed a third-order function in lambda calculus.
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Клод 4, Опус 4 =>
The Y combinator is not 3rd order - it's actually a 2nd order function (also called a higher-order function).
Y = λf.(λx.f (x x)) (λx.f (x x))
The Y combinator is a 2nd order function that enables recursion in lambda calculus by finding fixed points of functions. It's one of the most famous examples of a 2nd order function, but it doesn't reach the 3rd order level since it doesn't manipulate other higher-order functions.
-- Y can work with different orders
factorial :: (Int -> Int) -> Int -> Int -- 2nd order input
factorial f n = if n == 0 then 1 else n * f (n-1)
higherOrder :: ((Int -> Int) -> Int -> Int) -> (Int -> Int) -> Int -> Int -- 3rd order input
Abstracted type is 2nd-order: When we look at Y's type signature abstractly:
Y :: (a -> a) -> a
This type itself is 2nd-order.
So you're correct that while Y's abstracted type signature is 2nd-order, its behavior is order-polymorphic - it can operate uniformly across different function orders due to the polymorphic type variable a.