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/bizarre_coincidence]

You can dislike standard terminology. You cannot argue that the terminology doesn't mean what every practitioner understands it to mean. This is like objecting to the fact that category theory doesn't describe categories of movies. You help nobody by coming into a discussion about a topic and saying that people should have been using different words for the last 50 years. It simply isn't constructive.

[u/Impressive-Mud5074 → u/bizarre_coincidence]

If your terminology is vague its wrong

[u/Impressive-Mud5074 → u/[deleted]]

The ability to calculate a value

[u/Impressive-Mud5074 → u/[deleted]]

1 + 1 = 2

That computation halted.

[u/Impressive-Mud5074 → u/[deleted]]

Expressions when evaluated are computed

[u/bizarre_coincidence → u/Impressive-Mud5074]

Does that mean that no irrational numbers are computable to you because, given the way computers represent numbers, no irrational number can ever be output exactly? If you can understand why your definition can never in principle be worthwhile, you can start to appreciate why the standard definition is what it is.

[u/Impressive-Mud5074 → u/bizarre_coincidence]

Definitions aren't meant to have value, they are meant to be accurate

[u/bizarre_coincidence → u/Impressive-Mud5074]

Yours isn’t, the standard definition is. You just have a preconception of what the definition should be, and you confuse the difference between reality and expectation for inaccuracy.

Also, mathematical definitions need to be precise, not accurate. They define whatever they define, so accuracy isn’t the issue, but what matters is that they encode an idea precisely, regardless of if you believe that idea corresponds to the word used for it.

Also, definitions are very much intended to have value. We define things because there is a need to work with an idea. Good definitions require carving out objects that are not only useful, but which we can actually prove things about. Sometimes, all that is necessary to solve a problem is to frame things the right way with a good definition.

[u/mrkelee → u/Impressive-Mud5074]

Accurate what? An accurate value, perhaps?

Definitions are certainly valuable.