What I described isn't really an algorithm, it's just taking the digits of a number, let's say:

foo=3.14159265...

Where after 5 is some continuing sequence of decimals.

The series of functions is literally just:

foo(0) = 3 foo(1) = 3.1 foo(2) = 3.14...

And to be clear, it's not just like, an algorithm that estimates pi, it's literally just a list of return values that is infinitely long that return more and more digits of whatever the number is. That is actually how he defines pi.

https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080

pi _happens_ to be computable, and there are more efficient functions that will produce those numbers, but you could do the same thing with an incomputable number, you just need a definition for the number which is infinitely long.

To be clear, I don't think any of this is a good idea, just pointing out that if he's going to allow that kind of definition of pi (ie, admit a definition that is just an infinite list of decimal representations), you can just do the same thing with any real number you like. He of course will say that he's _not_ allowing any _infinite list_, only an arbitrary long one.

That's the key point though, this list isn't infinitely long, and all the numbers in it are rational. And it is an algorithm (specifically, a lookup table).

All the numbers you get this way are going to be rational, and if you require them to be finite, you can't even identify them with any irrational numbers. At least with the computable numbers you get an infinite set of irrational numbers along with the rationals, while still never touching the vast majority of all numbers (the remaining, incomputable irrationals).