Deciding how computations stop - The Halting Problem

• What procedures exist, if any, to determine whether a computation will stop ?
• Are there general computational procedures that allow us to ascertain things of this kind ?

The answer to these questions is found in a profound, fundamental and important theorem of mathematical logic called ``Gödel's Incompleteness Theorem". Roughly it asserted (proved) that

For our purposes this theorem tells us that any set of formalized rules will be insufficient to determine whether an arbitrary computation will halt.