We see arguments on morality and God sometimes have God observing and knowing the outcome of a decision. God is the ultimate voyeur.

OK, the theists will now be getting that tingling feeling so lets say you decided to choose an action according to if a Turing Machine halted.

For God to know what action you would take then God would have to know the solution to the halting problem.

The halting problem is an undecidable problem (i.e. it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer ). We've proven that one.

Obviously another way of doing this is to simply toss a die but we do get die-hard determinists so using the undecidable Halting problem allows me to eliminate determinism from the problem.

God doesn't know the outcome. You are making a decision that God is not privy to.

You still have the free-will to decide to have used the TM halting problem in the first place, but this ensures that unless God has some magic means of seeing all natural numbers, we have blinkered God to waiting for the machine to halt.

What flaws does this line of reasoning have ?.

There is no single algorithm that can be used to determine whether or not each conceivable Turing machine will halt, but that doesn't mean that there's no separate algorithm for each Turing machine.

Broiled Jogger wrote:There is no single algorithm that can be used to determine whether or not each conceivable Turing machine will halt, but that doesn't mean that there's no separate algorithm for each Turing machine.

We still confuse God; the program itself is arguably an algorithm but that is determined a posteriori by running the program. God thus has to run the TM with the program and wait for the result. Arguably God can do this very fast but the person with the free will doesn't have that luxury of timelessness. God thus make a decision that the time to halt is too long and that the person waiting will in fact give in. We thus constrain God to second guessing the waiting person to take a decision that they have not yet taken.

I can tell whether or not a sufficiently simple computer program will halt without running it. A vastly superior mind might be able to do so with with any program that our universe could produce.

Broiled Jogger wrote:I can tell whether or not a sufficiently simple computer program will halt without running it. A vastly superior mind might be able to do so with with any program that our universe could produce.

And a TM can do exactly what you are doing, or God for that matter. But you cannot decide that the program will halt unless you actually run the program logic and though this may be true for a sufficiently simple computer program I'm using the case of any and arbitrary program. For instance we could randomly create a program and run that and make our decision based on if that halted or not.

. See the Halting problem here.

The superior intelligence would not have to be able to tell whether or not any program at all would halt. It need only be concerned with programs we could create. All of them might be trivial to some sufficiently intelligent being.

Broiled Jogger wrote:The superior intelligence would not have to be able to tell whether or not any program at all would halt. It need only be concerned with programs we could create. All of them might be trivial to some sufficiently intelligent being.

But in the end it is proven that they can never decide if the program will halt. It doesn't matter how superior their intelligence is - they are fighting reality.

No one can create an algorithm that can tell if every program will halt, but we can't write every program. A sufficiently intelligent being might not have a problem with every program that we can write. In order to let the halting of a program determine your future action, you have to actually write and run the program.

You could solve the Entscheidungsproblem for an arbitrary string with an Oracle machine, if you could build one, which you can't. And you could solve it for all possible strings with a nondeterministic Oracle machine, if you could do that, which you can't. But people keep saying that God is infinite, and you only really need aleph-null to do the first and aleph-one to do the second (assuming the obvious exponential approach).

Still, you wouldn't be able to answer the continuum hypothesis without the axiom of choice. Or at least I don't think so. Still, if God could figure out a way to prove exactly how much a non-deterministic Oracle machine would overcount c, I'd love to hear it.

epepke wrote:You could solve the Entscheidungsproblem for an arbitrary string with an Oracle machine, if you could build one, which you can't. And you could solve it for all possible strings with a nondeterministic Oracle machine, if you could do that, which you can't. But people keep saying that God is infinite, and you only really need aleph-null to do the first and aleph-one to do the second (assuming the obvious exponential approach).

Still, you wouldn't be able to answer the continuum hypothesis without the axiom of choice. Or at least I don't think so. Still, if God could figure out a way to prove exactly how much a non-deterministic Oracle machine would overcount c, I'd love to hear it.

Yes, CH was Hilbert's first problem and we're stuck with it being impossible to prove or disprove within ZFC with or without an axiom of choice. What is odd is how so few religious texts present God in these terms and yet these terms are what would not so much as limit God but would actually define the scope of God in concrete terms that are culturally neutral.