God, Gödel, and Omniscience

[Nihilist Virus]

Omniscience is just as impossible as the classic "married bachelor."

Gödel's Incompleteness Theorem shows that any non-trivial self-consistent system must contain a logically valid statement which cannot be shown to be either true or false from the axioms.

Whether or not mathematics is consistent is currently an open question, but we already found a hypothesis which cannot be proven or disproven from our axioms: that there exists no X such that |Z|<|X|<|R| (Google "continuum hypothesis").

This presents a limitation on God's knowledge. God *cannot* know whether or not X exists. But that's ok, right, because theologians have already clarified that God is not omnipotent but rather maximally powerful (to solve the "Can God create a rock so big..." paradox). So in the same way, God is not omniscient but rather maximally knowledgeable, right?

Except... we have a huge problem. Not only does omniscience cause a paradox, but being maximally knowledgeable does as well. God's maximal knowledge would make him aware of every possible set; this exhaustive knowledge would force him to know whether or not X exists: if he knows of the set, then it exists; if he doesn't, then it doesn't exist. So he knows whether or not it exists, but he cannot know that: a paradox.

Now I know what you're thinking. "Silly Nihilist Virus and his human viral logic..." But guess what: until you can show God interacting with reality in any detectable way, he is nothing more than an idea... a human idea.

And some human ideas are better than others. But no human idea *ever* conceived is better than mathematics.

So we strip away at God and what's left? Omniscience crumbling to maximal knowledge crumbling even further to... what?... "lots" of knowledge? At what point does he stop being a God? At what point have we actually gone and proved a negative - that there is no God?

 

[2PhiloVoid]

Omniscience is just as impossible as the classic "married bachelor."

Gödel's Incompleteness Theorem shows that any non-trivial self-consistent system must contain a logically valid statement which cannot be shown to be either true or false from the axioms.

Whether or not mathematics is consistent is currently an open question, but we already found a hypothesis which cannot be proven or disproven from our axioms: that there exists no X such that |Z|<|X|<|R| (Google "continuum hypothesis").

This presents a limitation on God's knowledge. God *cannot* know whether or not X exists. But that's ok, right, because theologians have already clarified that God is not omnipotent but rather maximally powerful (to solve the "Can God create a rock so big..." paradox). So in the same way, God is not omniscient but rather maximally knowledgeable, right?

Except... we have a huge problem. Not only does omniscience cause a paradox, but being maximally knowledgeable does as well. God's maximal knowledge would make him aware of every possible set; this exhaustive knowledge would force him to know whether or not X exists: if he knows of the set, then it exists; if he doesn't, then it doesn't exist. So he knows whether or not it exists, but he cannot know that: a paradox.

Now I know what you're thinking. "Silly Nihilist Virus and his human viral logic..." But guess what: until you can show God interacting with reality in any detectable way, he is nothing more than an idea... a human idea.

And some human ideas are better than others. But no human idea *ever* conceived is better than mathematics.

So we strip away at God and what's left? Omniscience crumbling to maximal knowledge crumbling even further to... what?... "lots" of knowledge? At what point does he stop being a God? At what point have we actually gone and proved a negative - that there is no God?

At what point does mathematics stop being mathematics? OR...should I listen to Plato and throw away my Morris Kline book?

 

[Nihilist Virus]

Clarification:

Strictly speaking, Gödel's Incompleteness Theorem shows that X cannot be *shown* to either exist or not exist. So, conceivably, God could know whether X exists but simply not know the proof (since the proof does not exist).

But again this falls apart.

Suppose X does not exist. Then God, knowing all possible sets and their cardinality, could arrange all sets by their cardinality and thus show the gap in the continuum: that sets, arranged by cardinality, jump from being bijective with Z to being bijective with R. This would constitute an exhaustive proof that X does not exist.

Suppose X does exist. Then simply enumerating it constitutes proof of its existence.

The only other possibility is that there are certain infinite sets, X being one of them, which God cannot enumerate. But that should be impossible if he is maximally knowledgeable because he should know what every element of every set is, and such knowledge constitutes enumeration.

 

[zippy2006]

Omniscience is just as impossible as the classic "married bachelor."

Gödel's Incompleteness Theorem shows that any non-trivial self-consistent system must contain a logically valid statement which cannot be shown to be either true or false from the axioms.

Whether or not mathematics is consistent is currently an open question, but we already found a hypothesis which cannot be proven or disproven from our axioms: that there exists no X such that |Z|<|X|<|R| (Google "continuum hypothesis").

This presents a limitation on God's knowledge. God *cannot* know whether or not X exists. But that's ok, right, because theologians have already clarified that God is not omnipotent but rather maximally powerful (to solve the "Can God create a rock so big..." paradox). So in the same way, God is not omniscient but rather maximally knowledgeable, right?

Except... we have a huge problem. Not only does omniscience cause a paradox, but being maximally knowledgeable does as well. God's maximal knowledge would make him aware of every possible set; this exhaustive knowledge would force him to know whether or not X exists: if he knows of the set, then it exists; if he doesn't, then it doesn't exist. So he knows whether or not it exists, but he cannot know that: a paradox.

Now I know what you're thinking. "Silly Nihilist Virus and his human viral logic..." But guess what: until you can show God interacting with reality in any detectable way, he is nothing more than an idea... a human idea.

And some human ideas are better than others. But no human idea *ever* conceived is better than mathematics.

So we strip away at God and what's left? Omniscience crumbling to maximal knowledge crumbling even further to... what?... "lots" of knowledge? At what point does he stop being a God? At what point have we actually gone and proved a negative - that there is no God?

God's knowledge is not axiomatic, nor even rational in the strict sense--involving ratiocination. It is purely intellectual, like the angels, but even more than the angels, God knows everything that exists exhaustively due to the very fact that he has created everything that exists. He knows it immediately, without reasoning from point A to point B. Godel's theorems have to do with the human mode of reasoning whereby we move from premises to conclusions by a rational process. There is every reason to believe that such theorems do not apply to God (or even to angels).

 

[KevinSim]

This presents a limitation on God's knowledge. God *cannot* know whether or not X exists. But that's ok, right, because theologians have already clarified that God is not omnipotent but rather maximally powerful (to solve the "Can God create a rock so big..." paradox). So in the same way, God is not omniscient but rather maximally knowledgeable, right?

Except... we have a huge problem. Not only does omniscience cause a paradox, but being maximally knowledgeable does as well. God's maximal knowledge would make him aware of every possible set; this exhaustive knowledge would force him to know whether or not X exists: if he knows of the set, then it exists; if he doesn't, then it doesn't exist. So he knows whether or not it exists, but he cannot know that: a paradox.

Nihilist Virus, what do you mean when you use the phrase maximally knowledgeable?

I myself have no use for the idea of the omniscience of God. All God has to know, as far as I am concerned, is how to preserve forever some good things. Such a knowledge doesn't appear to me to have to include knowledge of every little detail that's ever going to occur in the universe.

 

[Nihilist Virus]

God's knowledge is not axiomatic, nor even rational in the strict sense--involving ratiocination. It is purely intellectual, like the angels, but even more than the angels, God knows everything that exists exhaustively due to the very fact that he has created everything that exists. He knows it immediately, without reasoning from point A to point B. Godel's theorems have to do with the human mode of reasoning whereby we move from premises to conclusions by a rational process. There is every reason to believe that such theorems do not apply to God (or even to angels).

Does God know whether X exists?

 

[KevinSim]

Does God know whether X exists?

I am pretty sure that God does not know whether or not X exists.

 

[KevinSim]

I am pretty sure that God does not know whether or not X exists.

Oops! I thought you were responding to me! I didn't realize you were responding to Zippy2006.

 

[Nihilist Virus]

Nihilist Virus, what do you mean when you use the phrase maximally knowledgeable?

Knowing the truth value of every possible proposition so long as such knowledge does not create a paradox.

I myself have no use for the idea of the omniscience of God. All God has to know, as far as I am concerned, is how to preserve forever some good things. Such a knowledge doesn't appear to me to have to include knowledge of every little detail that's ever going to occur in the universe.

When God says that "the hairs on your head are numbered," that's generally taken to mean that he does know every little detail that's ever going to occur in the universe. I'm showing that such an idea is false.

 

[Nihilist Virus]

I am pretty sure that God does not know whether or not X exists.

Do you accept, then, that God is not omniscient?

Oops! I thought you were responding to me! I didn't realize you were responding to Zippy2006.

You're certainly welcome to participate in the discussion. My response to him does not exclude you.

 

[zippy2006]

Does God know whether X exists?

Do you know that there are two Incompleteness Theorems from Godel, and thus "Godel's Incompleteness Theorem" provides no accurate referent?

Do you know that the first Incompleteness Theorem has nothing to do with the existence of entities, but has rather to do with the provability of certain truths contained in the language of the relevant system?

God knows whether everything exists, and it has nothing to do with axiomatic, or even mathematical, logic.

 

[Nihilist Virus]

Do you know that there are two Incompleteness Theorem's from Godel, and thus "Godel's Incompleteness Theorem" provides no accurate referent?

I said that it "shows that any non-trivial self-consistent system must contain a logically valid statement which cannot be shown to be either true or false from the axioms." How does that not clarify for you which one I'm referring to?

Do you know that the first Incompleteness Theorem has nothing to do with the existence of entities, but has rather to do with the provability of certain truths contained in the language of the relevant system?

Do you know that showing the existence of something constitutes proof of the claim that such a thing exists? Did you miss the part where I referred to the continuum hypothesis?

God knows whether everything exists, and it has nothing to do with axiomatic, or even mathematical, logic.

Once again, either X exists or it doesn't. If it does, and if God "knows whether everything exists" then he knows every element of the set and hence he can enumerate it, proving its existence. But he cannot prove its existence as shown in the analysis of the continuum hypothesis, so X must not exist. But if X does not exist, and if God "knows whether everything exists" then he can arrange all sets by their cardinality and demonstrate the gap in the continuum, constituting an exhaustive proof that X does not exist. But again, he cannot prove that.

This is a paradox resulting from omniscience. You shouldn't be surprised... the other omni-properties of God also result in paradoxes.

But as I said in the OP, God must not only be reduced to maximal knowledge but there must be some quantifiable sets that he is not aware of. Otherwise, being aware of all possible sets results in the above paradox. Thus, God is not even maximally knowledgeable.

I'm aware that you want to say that God has no need for logic since logic is used in discovering that which was previously unknown, but the fact that God has no need of logic does not cripple him from using it. He should still be able to enumerate sets and thus the paradox is unavoidable.

 

[KevinSim]

Knowing the truth value of every possible proposition so long as such knowledge does not create a paradox.

Then I don't believe that God has maximal knowledge.

When God says that "the hairs on your head are numbered," that's generally taken to mean that he does know every little detail that's ever going to occur in the universe. I'm showing that such an idea is false.

I think it means that God understands each human soul intimately, now. I'm not convinced God knows the position of every grain of sand in that world, nor do I think God knows what every individual soul is going to do nine days from now.

 

[KevinSim]

Do you accept, then, that God is not omniscient?

Yes, absolutely.

You're certainly welcome to participate in the discussion. My response to him does not exclude you.

Thanks!

 

[zippy2006]

Once again, either X exists or it doesn't. If it does, and if God "knows whether everything exists" then he knows every element of the set and hence he can enumerate it, proving its existence. But he cannot prove its existence as shown in the analysis of the continuum hypothesis, so X must not exist. But if X does not exist, and if God "knows whether everything exists" then he can arrange all sets by their cardinality and demonstrate the gap in the continuum, constituting an exhaustive proof that X does not exist. But again, he cannot prove that.

If CH is false then God could enumerate the intermediate set. If CH is true then God could demonstrate the gap. Why think he could not do these things?

And what of Skolem's Paradox?

Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by Skolem, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as Skolem's paradox, and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171).​

This is a paradox resulting from omniscience. You shouldn't be surprised... the other omni-properties of God also result in paradoxes.

Such as?

I'm aware that you want to say that God has no need for logic since logic is used in discovering that which was previously unknown, but the fact that God has no need of logic does not cripple him from using it. He should still be able to enumerate sets and thus the paradox is unavoidable.

Either CH is (for God) provably true, provably false, or unprovable. The first two possibilities pose no problems, albeit humans have not yet had success in either direction. Given something like Skolem's Paradox I see no reason why the third possibility would pose a problem.

 

[Nihilist Virus]

Then I don't believe that God has maximal knowledge.


I think it means that God understands each human soul intimately, now. I'm not convinced God knows the position of every grain of sand in that world, nor do I think God knows what every individual soul is going to do nine days from now.

Is this idea heretical?

 

[Nihilist Virus]

If CH is false then God could enumerate the intermediate set. If CH is true then God could demonstrate the gap. Why think he could not do these things?

The continuum hypothesis was proved to be undecidable. Doing those things would prove that X does or does not exist, which would answer the continuum hypothesis.

And what of Skolem's Paradox?

Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by Skolem, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as Skolem's paradox, and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171).​

I don't understand the first sentence. Sets are primitive objects and as such are undefined. That issue seems to be metamathematics I guess.

Such as?

You don't seem to have read the OP.

Either CH is (for God) provably true, provably false, or unprovable. The first two possibilities pose no problems, albeit humans have not yet had success in either direction. Given something like Skolem's Paradox I see no reason why the third possibility would pose a problem.

It's Gödel's Incompleteness Theorem that shows the third option is the case here. I appreciate the research you put into this issue, but maybe try re-reading the OP.

And if Skolem's paradox resolves this, please dumb it down for me.

 

[2PhiloVoid]

Omniscience is just as impossible as the classic "married bachelor."

Gödel's Incompleteness Theorem shows that any non-trivial self-consistent system must contain a logically valid statement which cannot be shown to be either true or false from the axioms.

Whether or not mathematics is consistent is currently an open question, but we already found a hypothesis which cannot be proven or disproven from our axioms: that there exists no X such that |Z|<|X|<|R| (Google "continuum hypothesis").

This presents a limitation on God's knowledge. God *cannot* know whether or not X exists. But that's ok, right, because theologians have already clarified that God is not omnipotent but rather maximally powerful (to solve the "Can God create a rock so big..." paradox). So in the same way, God is not omniscient but rather maximally knowledgeable, right?

Except... we have a huge problem. Not only does omniscience cause a paradox, but being maximally knowledgeable does as well. God's maximal knowledge would make him aware of every possible set; this exhaustive knowledge would force him to know whether or not X exists: if he knows of the set, then it exists; if he doesn't, then it doesn't exist. So he knows whether or not it exists, but he cannot know that: a paradox.

Now I know what you're thinking. "Silly Nihilist Virus and his human viral logic..." But guess what: until you can show God interacting with reality in any detectable way, he is nothing more than an idea... a human idea.

And some human ideas are better than others. But no human idea *ever* conceived is better than mathematics.

So we strip away at God and what's left? Omniscience crumbling to maximal knowledge crumbling even further to... what?... "lots" of knowledge? At what point does he stop being a God? At what point have we actually gone and proved a negative - that there is no God?

Funny how none of this seemed to have killed the prospect of at least some religious cogency for Gödel. ...but it does for you, NV.

Clarification:

Strictly speaking, Gödel's Incompleteness Theorem shows that X cannot be *shown* to either exist or not exist. So, conceivably, God could know whether X exists but simply not know the proof (since the proof does not exist).

But again this falls apart.

Suppose X does not exist. Then God, knowing all possible sets and their cardinality, could arrange all sets by their cardinality and thus show the gap in the continuum: that sets, arranged by cardinality, jump from being bijective with Z to being bijective with R. This would constitute an exhaustive proof that X does not exist.

Suppose X does exist. Then simply enumerating it constitutes proof of its existence.

The only other possibility is that there are certain infinite sets, X being one of them, which God cannot enumerate. But that should be impossible if he is maximally knowledgeable because he should know what every element of every set is, and such knowledge constitutes enumeration.

...or it could just be that when mathematics get "too large" and encumbered by measures without pragmatic meaning on a human level, the significance of our axioms and systems "implodes."

I have a hard time seeing that all of this shows God would be at fault in His "knowledge" rather than that it simply illustrates that our maths are "limited" and faulty (or uncertain) on a human scale, as Kline (1980, 2011) seems to indicate in his assessments as to the state of modern mathematics.

Moreover, it seems that if we are going to focus on the work of Gödel, there are some other implications and applications his work might also have that should probably be taken into account as well, such as is suggested by David Goldman:
The God of the Mathematicians | David P. Goldman

References​

Goldman, D. P. (2010). THE GOD OF THE MATHEMATICIANS. First Things, (205), 45. Found online (as above).

Kline, Morris. (1980, 2011). Mathematics: The loss of certainty. New York, NY: Fall River Press.

 

[zippy2006]

The continuum hypothesis was proved to be undecidable. Doing those things would prove that X does or does not exist, which would answer the continuum hypothesis.

Rather, Godel showed that CH is consistent with ZFC set theory, and therefore that it could not be disproved in that system. Additionally, "Gödel believed that CH is false and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets." This seems to be something very similar to what Skolem argued.

So CH could be disproved by adding a true and accepted axiom to ZFC set theory, an axiom which disproves CH. Godel apparently believed that such an addition would be necessary to "adequately characterize the universe of sets," and Skolem apparently believed that something more than general set theory is required to determine the truth value of CH.

I don't understand the first sentence. Sets are primitive objects and as such are undefined. That issue seems to be metamathematics I guess.

According to Enterton's A Mathematical Introduction to Logic, a set "is a collection of things, called its members or elements." It goes on to give the basics of set theory, such as emptiness, union, intersection, disjointness, mapping, correspondence, etc. Apparently Skolem's claim is set theory does not provide the necessary machinery to properly answer finer questions about cardinality.

You don't seem to have read the OP.

Rather, I distinguish between an assertion and a conclusion, and ask for the syllogism. God is omnipotent, which means that he can do everything but that which is logically impossible. Since every rock is of finite mass, and God has infinite power, it is logically impossible for any rock to be such that God could not "lift" it.

It's Gödel's Incompleteness Theorem that shows the third option is the case here. I appreciate the research you put into this issue, but maybe try re-reading the OP.

And if Skolem's paradox resolves this, please dumb it down for me.

All Godel's theorem shows is that, given a certain set of axioms, CH cannot be disproven. Paul Cohen showed that it cannot be proven given these axioms. As Godel and Skolem both believed, this result may simply be the consequence of an insufficient set of axioms.

 

[Nihilist Virus]

Funny how none of this seemed to have killed the prospect of at least some religious cogency for Gödel. ...but it does for you, NV.

I'd doubt that Gödel believed God to be omnipotent.

...or it could just be that when mathematics get "too large" and encumbered by measures without pragmatic meaning on a human level, the significance of our axioms and systems "implodes."

I cannot discern meaning from that.

I have a hard time seeing that all of this shows God would be at fault in His "knowledge" rather than that it simply illustrates that our maths are "limited" and faulty (or uncertain) on a human scale, as Kline (1980, 2011) seems to indicate in his assessments as to the state of modern mathematics.

See above.

Moreover, it seems that if we are going to focus on the work of Gödel, there are some other implications and applications his work might also have that should probably be taken into account as well, such as is suggested by David Goldman:
The God of the Mathematicians | David P. Goldman

References​

Goldman, D. P. (2010). THE GOD OF THE MATHEMATICIANS. First Things, (205), 45. Found online (as above).

Kline, Morris. (1980, 2011). Mathematics: The loss of certainty. New York, NY: Fall River Press.

I'm not obsessed with Gödel. I'm just referencing a theorem.