Ever since I learned about them I found the Surreals a far more natural construction than the more classical one shown here. They don't just stop the logical induction at arbitrary points.
Unreal numbers
31–32 of 32 posts
Re: Unreal numbers
#32> But what would be an example of an uncomputable number? That’s a good question. Most obviously, we could be talking about numbers that encode the solution to the halting problem. It would lead to a paradox to have a computer program that allows us to decide, in the general case, whether a given computer program halts. So, if a procedure to approximate a particular real requires solving the halting problem, we can’t…
Enumerate all well formed programs in order. For programs that halt assign the digit 0, and for the ones that don't, the digit 1. Put the digits after a decimal point and interpret in binary.
Very interesting! Thank you for explaining this.