So, this is interesting. You say here that it is possible omniscience is a halting oracle, but then argue in this thread that halting oracles are a logical impossibility. Traditionally, people have thought that even an omniscient being like God cannot accomplish the logically impossible. Do you think there is something different at play here with regards to halting oracles?
Yes, can God make a rock so heavy that even He couldn’t lift it?
This is getting to metaphysics outside my training. The key thing for halting oracle is to be able to observe a true infinity. That is the challenge. The contradiction arises because the halting oracle, also, is a program subject to inclusion in a program itself.
In the case of God, both these two things might be side stepped. Perhaps God can create an infinite timeline outside himself, and observe all of it from outside that timeline. If that is the case, He could be a halting oracle without logical contradiction. Not that the halting oracle here is not subject to logical contradiction, because it can’t be called recursively as is required for the proof against halting oracles above. That is my sloppy attempt to show why God is not necessarily subject to the halting oracle logical contradiction.
Another solution could be that he just knows, without process or mechanism of knowing, he just knows. It is God after all, why not?
If God makes a rock so large no one else can lift it, and then He promises not to move it, than that rock is immovable to everyone including Him. Right?
This is more convincing to me. It could be that God’s omniscience is not a total computable function. It may be the case that God never has to compute anything - He just knows a (potential/actual?) infinite number of statements.
Well He supposedly knows everyone’s thoughts, so why not?
I think there is a theological debate on this one…though I am not sure where it stands.
I don’t think I’ve ever read a serious discussion of God’s omniscience in relation to halting oracles and information theory…I think it is a project that a Christian informed in theology and CS should take on.
Well, there is a relationship between the halting problem and incompleteness, so perhaps an opportunity to read something along these lines will present itself shortly (for certain values of ‘serious’).
One question that has interested me about halting oracles and God’s omniscience: Does traditional everlasting life require God to be a halting oracle? In other words, to create a world that will run forever, does God need to know in advance that it will, in fact, run forever in a halting sense? Possible complication - we don’t know for sure that the behavior of this universe is computable in the Turing sense, and so analysis with respect to Turing machines and halting may not apply.
Just remind people what a Halting Oracle is and how it is a logical impossibility:
Which gets to one divine loophole.
If God is more than a computational machine (which seems to be a very reasonable supposition) He cannot be called recursively in a way that creates the contradiction. So perhaps that one loophole that creates space for him to be halting oracle.
To the same extent entangled particles do not behave “rationally” omniscience doesmt have to behave rationally.
There’s a difference between rationality and logical possibility. Nothing about entanglement or quantum mechanics violates logic. But a halting oracle ostensibly results in a logical contradiction. Is it possible for God to overcome logical contradictions? I personally don’t think so. For example, I don’t think God can make squared circles, nor can He exist and not exist at the same time.
The question is subtly different. The logical contradiction is contextual. In a divine context, a different context than the original proof, is it still illogical?
I am not sure we know.