  The paper explores known results related to the problem of identifying if a
given program terminates on all inputs -- this is a simple generalization of
the halting problem. We will see how this problem is related and the notion of
proof verifiers. We also see how verifying if a program is terminating involves
reasoning through a tower of axiomatic theories -- such a tower of theories is
known as Turing progressions and was first studied by Alan Turing in the
1930's. We will see that this process has a natural connection to ordinal
numbers. The paper is presented from the perspective of a non-expert in the
field of logic and proof theory.
