Here's an example of a Python program whose halting behavior -- if run on a hypothetical Python interpreter with infinite RAM and the ability to make use of it

Here's an example of a Python program whose halting behavior -- if run on a hypothetical Python interpreter with infinite RAM and the ability to make use of it -- is currently unknown. It looks for counterexamples to the 3n+1 conjecture, and it tries every integer to see if it's a counterexample. start = i = 1 visited = [] while True: if i in visited: if 1 not in visited: break else: print(start, visited) visited = [] start += 1 i = start continue visited.append(i) if i % 2: i = 3*i + 1 else: i //= 2 On an actual computer's Python interpreter, I believe this will eventually halt with a MemoryError exception (although I'm not at all sure that you can actually run it long enough to hit that, as it will probably take many years!), failing to resolve the mathematical question. That potential for the MemoryError is why it matters whether your model has infinite memory or not! (You can make a considerably more memory-efficient version of this which should be able to search much further before hitting a MemoryError, but I don't expect it changes the ultimate fate of the program when run on a real physical computer.)