I don't know what you mean by this, and I'm not sure how it relates to what I said. I'm using "decidable" in the strict, computer-science sense of the word.
The statement in example 1 of the paper, which we're discussing, is about computable real numbers. A computable real number is one for which there is a Turing machine that can calculate it to any desired finite precision in finite time.
A semidecidable problem is one for which there is a program that halts if the answer is "yes", but may not halt if the answer is "no". The halting problem is in this category.
Given a computable real number x, asking whether x<0 or x>0 are semidecidable but not decidable problems.
Computability theory is mathematics.
ekez: why is it impossible to decide whether x<0, x=0, or x>0 as in Example 1?
lisper: Only x=0 is undecidable. It's because you have to check an infinite number of digits past the decimal point to see if all of them are zero, and that will not halt.
teraflop: That cannot be true, because if both x<0 and x>0 are decidable then x=0 is also decidable.
Both lisper (me) and teraflop are correct. The problem is that inequality is not decidable, it's semi-decidable. It's teraflop who introduced the word "decidable" into the discussion, not me, but I didn't pay enough attention to that at that time.
/u/scarmig got it right in this comment: https://news.ycombinator.com/item?id=41150519
Here is a concrete example: write a program that systematically tests all numbers for counterexamples to the Goldback conjecture. Every time a number fails to be a counterexample, output a zero, otherwise output a 1. Consider the output of that program to be a real number expressed as a binary decimal. If that number is not zero you will know in a finite amount of time, but if it is zero you will not know that until someone proves the Goldbach conjecture.