OK. What does “omniscience” mean? The root words translate to something like “all knowing”. But what is “all”, and what is “knowing”? What’s the minimum qualification? Each successive option seems harder to prove:
Option A: (sufficiently advanced aliens) God’s knowledge isn’t infinite or anything, just far beyond our current level. “Omniscience” is more metaphorical than literal.
Option B: (semi-omniscient simulator) God can look up any past event in the world simulation, but isn’t simultaneously conscious of all of them and cannot predict the future short of actually simulating it. He does not know all the logical implications of His knowledge and can be surprised by events. (Janet from The Good Place might be at this level.) Although perhaps he can rewind the simulation and try a different timeline, if He makes any changes, He can’t always predict what would happen without actually trying it. He may also be ignorant of events in His native plane, outside of the world simulation.
Option C: (halting oracle of the first degree) God is a halting oracle machine able to solve the halting problem for any Turing machine, but is unable to solve the halting problem for halting oracle machines like Himself.
Option D: (higher-order halting oracle) God is a halting oracle machine able to solve the halting problem for any Turing machine, and halting oracle machines of some finite degree less than His own, but is unable to solve the halting problem for higher-order halting oracle machines like Himself, or those of any higher degree. There may possibly be beings of greater degree that know things God doesn’t.
Options A, and maybe B seem at least possible, but very very far from proven. Option C seems unprovable using any finite amount of evidence, but probably has a logically coherent definition. Option D seems unprovable even with infinite evidence, but again seems coherent.
Or did you have some other option in mind? I don’t know how to get past Option D without self-referential paradoxes invalidating the whole definition, but perhaps you have some new math for me?
Yes, I have option E: Everything. God just know everything, all the possible universes, - not calculating, just having them in His memory that is infinite.
As I stated in the previous comment , there is no reason for the exact theory to be finite, while approximations can be finite (would you like me to copy it or you can find it?).
That’s your crux? Lesser interpretations than E won’t do?
I am not convinced that E is logically coherent. It’s as meaningless as “married bachelor”.
Suppose that God’s memory is the set of “all facts” O.
The set of all subsets (or powerset) of O, we’ll call p(O).
Then, for any given fact f, there is a further fact f ′ stating that it’s either in or not in each subset of O in p(O).
Thus, there must be at least as many facts as there are elements of p(O), which, being the powerset of O, by Cantor’s Theorem must have a strictly greater cardinality than O.
But we assumed that O contains all facts. Contradiction!
And Cantor’s Theorem holds even for infinite sets! Q.E.D.
Good evening. Sorry to bring up this old thread. Your discussion was very interesting. Specifically regarding this comment, one thing confuses me. Isn’t “the memory of an omniscient God” in this thought experiment the same as “the set of all existing objects in all existing worlds”? If your reasoning about the set paradox proves that “the memory of an omniscient God” cannot exist, doesn’t that prove that “an infinite universe” cannot exist either? Or is there a difference between the two? (Incidentally, I would like to point out that the universe and even the multiverse can be finite. Then an omniscient monotheistic God would not necessarily have infinite complexity. But for some reason many people forget this.)
Well, your argument should be able to kill the concept of Tegmark mathematical multiverse then, so you can guess it is not a “silver bullet” :) Two possible answers:
1. You can not just change the word “mathematical universe” to the word “fact” in my definition E. “Mathematical universe stating that...” makes no sense for me.
However, there are different set theory axiomatics. Some of them allow universal sets
OK, that’s a good point. I had not heard of the universal sets that contain themselves, which I thought would lead to contradictions.
should be able to kill the concept of Tegmark mathematical multiverse
I’m really not persuaded by the MUH, but at least it’s based on reasoned a priori arguments. Do you have similar a priori arguments for God? There’s no way for evidence to ever be enough establish omniscience by itself.
Yeah, given New Foundations, I’m no longer confident that “omniscience” is a logical contradiction, but neither am I confident that it isn’t. And I still think it would take an infinite amount of evidence to prove inductively, so you would need some kind of a priori argument for it instead (or why believe it at all?). That’s one obstacle down, but still a long way to go.
OK. What does “omniscience” mean? The root words translate to something like “all knowing”. But what is “all”, and what is “knowing”? What’s the minimum qualification? Each successive option seems harder to prove:
Option A: (sufficiently advanced aliens) God’s knowledge isn’t infinite or anything, just far beyond our current level. “Omniscience” is more metaphorical than literal.
Option B: (semi-omniscient simulator) God can look up any past event in the world simulation, but isn’t simultaneously conscious of all of them and cannot predict the future short of actually simulating it. He does not know all the logical implications of His knowledge and can be surprised by events. (Janet from The Good Place might be at this level.) Although perhaps he can rewind the simulation and try a different timeline, if He makes any changes, He can’t always predict what would happen without actually trying it. He may also be ignorant of events in His native plane, outside of the world simulation.
Option C: (halting oracle of the first degree) God is a halting oracle machine able to solve the halting problem for any Turing machine, but is unable to solve the halting problem for halting oracle machines like Himself.
Option D: (higher-order halting oracle) God is a halting oracle machine able to solve the halting problem for any Turing machine, and halting oracle machines of some finite degree less than His own, but is unable to solve the halting problem for higher-order halting oracle machines like Himself, or those of any higher degree. There may possibly be beings of greater degree that know things God doesn’t.
Options A, and maybe B seem at least possible, but very very far from proven. Option C seems unprovable using any finite amount of evidence, but probably has a logically coherent definition. Option D seems unprovable even with infinite evidence, but again seems coherent.
Or did you have some other option in mind? I don’t know how to get past Option D without self-referential paradoxes invalidating the whole definition, but perhaps you have some new math for me?
Yes, I have option E: Everything. God just know everything, all the possible universes, - not calculating, just having them in His memory that is infinite.
As I stated in the previous comment , there is no reason for the exact theory to be finite, while approximations can be finite (would you like me to copy it or you can find it?).
That’s your crux? Lesser interpretations than E won’t do?
I am not convinced that E is logically coherent. It’s as meaningless as “married bachelor”.
Suppose that God’s memory is the set of “all facts” O.
The set of all subsets (or powerset) of O, we’ll call p(O).
Then, for any given fact f, there is a further fact f ′ stating that it’s either in or not in each subset of O in p(O).
Thus, there must be at least as many facts as there are elements of p(O), which, being the powerset of O, by Cantor’s Theorem must have a strictly greater cardinality than O.
But we assumed that O contains all facts. Contradiction!
And Cantor’s Theorem holds even for infinite sets! Q.E.D.
Did I just disprove God?
Good evening. Sorry to bring up this old thread. Your discussion was very interesting. Specifically regarding this comment, one thing confuses me. Isn’t “the memory of an omniscient God” in this thought experiment the same as “the set of all existing objects in all existing worlds”? If your reasoning about the set paradox proves that “the memory of an omniscient God” cannot exist, doesn’t that prove that “an infinite universe” cannot exist either? Or is there a difference between the two? (Incidentally, I would like to point out that the universe and even the multiverse can be finite. Then an omniscient monotheistic God would not necessarily have infinite complexity. But for some reason many people forget this.)
Well, your argument should be able to kill the concept of Tegmark mathematical multiverse then, so you can guess it is not a “silver bullet” :) Two possible answers:
1. You can not just change the word “mathematical universe” to the word “fact” in my definition E. “Mathematical universe stating that...” makes no sense for me.
2. Cantor’s theorem is based on particular set axiomatic. However, there are different set theory axiomatics. Some of them allow universal sets https://en.wikipedia.org/wiki/Universal_set
OK, that’s a good point. I had not heard of the universal sets that contain themselves, which I thought would lead to contradictions.
I’m really not persuaded by the MUH, but at least it’s based on reasoned a priori arguments. Do you have similar a priori arguments for God? There’s no way for evidence to ever be enough establish omniscience by itself.
″ OK, that’s a good point. I had not heard of the universal sets than contain themselves, which I thought would lead to contradictions. ”
Great, the update of belief :)
Yeah, given New Foundations, I’m no longer confident that “omniscience” is a logical contradiction, but neither am I confident that it isn’t. And I still think it would take an infinite amount of evidence to prove inductively, so you would need some kind of a priori argument for it instead (or why believe it at all?). That’s one obstacle down, but still a long way to go.