Q:

Why is the halting problem unsolvable by a turing machine?

So my knowledge of CS is amateurish at best but to me, logically, it seems like the halting problem is solvable.
So any human can determine if a problem halts with rigorous inspection, so why can't a very advanced, say human AI, solve the halting problem?
Also, is there a machine, say more powerful than a turing machine, that can solve this problem?
PS: I might be making some bad assumptions, please point that out if so.

A:

It can be proved that the halting problem cannot be solved by any Turing machine. There are several question regarding this on this side which you can look up.
It is not true that humans can determine if a given Turing machine halts. For example, we have been unable so far to compute many values of busy beaver functions, which measure the maximal number of steps a halting Turing machine can run given the number of tape symbols and number of states.
According to the Church–Turing thesis, every machine can be simulated by a Turing machine. So DNA computers, quantum computers and the like will not help. All they can do (especially the latter) is to dramatically speed up computation, but they cannot help computing non-computable functions.
There is a field known as hyper-computation which studies machines stronger than Turing machines, but these are not believed to be realizable by most practitioners. Recursion theory explores such models from a more theoretical perspective.

A:

The major bad assumption is that "any human can determine if a problem halts with rigorous inspection". The strong Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. So consider the following program.
n = 4
loop
  if there exists two primes p and q and p + q = n then
    n = n + 2
  else
    return
  endif
end loop

The program loops until it finds an even integer that can't be expressed as the sum of two primes. So if it halts, the conjecture is proven false. If indeed "any human could determine if a program halts", we could settle this, and several other famous conjectures (at least theoretically).

