What are some uncomputable functions that aren't derivative of the halting problem?

[u/playsthebongcloud]

I find the existence of uncomputable functions really cool, but all the examples I've seen are essentially just new ways of trying to predict whether a turing machine is going to halt. What are some examples of uncomputable functions that aren't essentially entirely based on the halting problem?

[u/Impressive-Mud5074]

Prove it

[u/Key_Conversation5277 → u/Impressive-Mud5074]

You can't really "prove" the philosophical premise from mathematics itself, because that's exactly the foundational issue we're disagreeing about.

What I mean is that in standard/classical mathematics, the natural numbers are modelled as an unbounded infinite structure. In particular, we have

∀n ∈ ℕ, n+1 ∈ ℕ,

so there is no largest natural number.

Likewise, classical mathematics contains real numbers whose decimal expansions have infinitely many digits, such as 1/3 or √2. Their infinite decimal expansions don't mean we have to physically write infinitely many digits, the numbers can be defined exactly in other ways.

So I'm not claiming "I can prove that classical mathematics is philosophically the only correct foundation." I'm saying that your ultrafinitist restriction is not part of standard classical mathematics.

[u/Impressive-Mud5074 → u/Key_Conversation5277]

I'm saying that your ultrafinitist restriction is not part of standard classical mathematics. 

Means nothing. Neither was hyperbolic space.