The Entscheidungsproblem (German for decision problem) is a challenge posed by David Hilbert. The problem requires an algorithm that, given all the axioms of math, can determine whether a mathematical statement can be proven.
The implications of the Halting Problem applies to automatic program verification, since most programs can be reduced to the Halting Problem.
Proof Of Undecidability
This problem was given a negative answer (a proof that it is impossible/undecidable) by Alan Turing by using Turing Machines. Since Turing stated that every computable statement can be run on a Turing Machine, he rephrased the Halting Problem:
Given a program, can we determine if it halts, or runs forever
This is the exact same as the decision problem, which states
Can a given mathematical statement be proved?
They both refer to the decidability of the problem. If a problem is decidable, it means there is an answer for it, either True or False.
A summary by Tom Scott on YouTube: