We often extend arithmetic with new sorts of numbers to solve previously unsolvable equations like sqrt(-1) - is finding a fixed point of f(x) = x + 1 in this tradition? Does it point to an extension of arithmetic?

Also I find this confusing:

"Applied to a function with one variable, the Y combinator usually does not terminate"

What does it mean for a function to terminate? Is it like the limit of a series?

> Does it point to an extension of arithmetic?

Well, the fixed point of f(x)=x+1 can be interpreted as infinity. So kind of yes?

> What does it mean for a function to terminate?

Lambda calculus is a formal system that describes computation. So it literally means termination like when a computer program finishes running / halts. You have text which describes your program and you apply a few rewriting rules until you can’t. If at some point you can’t apply any more rewriting rules, you’ve reached a final state, your program has terminated.

I suppose, when you apply Y combinator eg to f(x)=x+1 then you are just going to get x+1+1+… and so on as you keep applying rewriting rules.