Sorry if this belongs on /g/. According to the YouTube vid, https://youtu.be/RcVA8Nj6HEo?si=Km1_ngxCfakjZ7nX
Around 19 minutes in, the YouTuber starts talking about fixed points of lambda terms and real valued functions. The fixed point of x^2+2 only has fixed points in the complex numbers. My question is, will applying the fixed point combinator to the lambda term for x^2+2 result in a lambda term that "acts like" a complex number?

idk, but if you can make that thing act as a 2x2 matrix you can make it walk&talk like a complex number
https://en.wikipedia.org/wiki/Matrix_(mathematics)#Applications

>>16699646
You can, but I'm talking specifically about the encoding built off of the naturals in the video.

>>16699647
Yes, you can do a bunch of computing BS to make anything into anything. The cross diagram approach he chose is retarded.

>>16699607 (OP)
The fixed point combinator he gives is in untyped lambda calculus, and is a special case of Klop's (https://www.cs.ru.nl/barendregt60/essays/klop/art16_klop.pdf) for n-1=1 binders.
The reduct will just 'act like' another term (because of the unityping), or an infinite loop if full normalization is non-terminating. You need a stronger type system to have any hope of formalizing "acts like" in greater depth.

>>16700014
Sorry if I'm unclear, or just too dumb to understand. What I meant by "acts like" is that I would be able to apply arithmetic operations on it with the naturals that have already been defined, and be able to get back to a natural. Since the fixed points in complex numbers of x^2 +2x+2 are (-1+-iroot(6))/2, and multiply 2, add 1, and squaring have already been defined, can I perform those operations, and then square again to get the lambda term for 36 that is consistent with the recursive definition of the naturals that has already been defined? In this way, the fixed point combinator would work like some kind of field extender.

>>16700014
>>16700048
After reading the paper, I think I can answer my own question. Since there are infinitely many unique fixed point combinators, and only finitely many fixed points of a parabola, there's no guarantee that the fixed point combinator I apply to my parabola will be the one that results in something that will act like a proper complex number with the previously defined operations. I still wonder if it's possible to find such a combinator.