With a assumtion of unlimited time even a computer can decide the halting problem by just running the program in question to test if it halts. The issue is that the task is to determine for ALL programs if they halt and for each of them to determine that in a FINITE amount of time.
> What makes you certain there is an algorithmic analog for all human reasoning?
(Maybe) not for ALL human thought but at least all communicatable deductive reasoning can be encoded in formal logic.
If I give you an algorithm and ask you to decide if it does halt or does not halt (I give you plenty of time to decide) and then ask you to explain to me your result and convince me that you are correct, you have to put your thoughts into words that I can understand and and the logic of your reasoning has to be sound. And if you can explain to me you could as well encode your though process into an algorithm or a formal logic expression. If you can not, you could not convince me. If you can: now you have your algorithm for deciding the halting problem.
> What makes you certain there is an algorithmic analog for all human reasoning?
(Maybe) not for ALL human thought but at least all communicatable deductive reasoning can be encoded in formal logic. If I give you an algorithm and ask you to decide if it does halt or does not halt (I give you plenty of time to decide) and then ask you to explain to me your result and convince me that you are correct, you have to put your thoughts into words that I can understand and and the logic of your reasoning has to be sound. And if you can explain to me you could as well encode your though process into an algorithm or a formal logic expression. If you can not, you could not convince me. If you can: now you have your algorithm for deciding the halting problem.