The Big Rock Paradox: Stephen Hawking's Thought-Experiment

[Karl G.]

I was first introduced to this thought-experiment upon reading Stephen Hawking's A Brief History of Time . Suppose an omnipotent being exists. If it does, it would be able to do anything (by definition!). Therefore, it would be able to produce a rock it couldn't lift. Therefore, it wouldn't be able to do anything it wants, therefore it wouldn't be omnipotent. What do you think?
Please nothe that I have phrased this thread to avoid it being locked. Please keep that in mind if you comment.

 

[JoeDawg]

I was first introduced to this thought-experiment upon reading Stephen Hawking's A Brief History of Time . Suppose an omnipotent being exists. If it does, it would be able to do anything (by definition!). Therefore, it would be able to produce a rock it couldn't lift. Therefore, it wouldn't be able to do anything it wants, therefore it wouldn't be omnipotent. What do you think?
Please nothe that I have phrased this thread to avoid it being locked. Please keep that in mind if you comment.

An all powerful being, would be capable of making itself as 'physically strong', at any given moment, as it wanted to. So it makes itself physically capable of lifting a certain amount. Then it makes a rock that weighs more. Now it can't lift it. Then it decides it really wants to, so it makes itself physically stronger. Voila.

 

[Karl G.]

Hmmm... your argument is invalid due to the definition of omnipotence. An omnipotent being can't make itself stronger to lift a rock ... its omnipotence already ensures it can do that.

 

[Ivan Seeking]

I prefer the other version of this question: Could God make a salsa so hot that he can't eat it?

 

[JoeDawg]

Hmmm... your argument is invalid due to the definition of omnipotence. An omnipotent being can't make itself stronger to lift a rock ... its omnipotence already ensures it can do that.

It ensures that it 'can', not that it 'must' in any particular instance.

 

[DaveC426913]

To me, this has little to do with a thought-experiment and much more to do with the semantics of "omnipotent".

i.e. so there's a word in the English language wherein you can form sentences that are seemingly paradoxical. So what?

 

[Karl G.]

To me, this has little to do with a thought-experiment and much more to do with the semantics of "omnipotent".

i.e. so there's a word in the English language wherein you can form sentences that are seemingly paradoxical. So what?

The fact that 'omnipotent' is paradoxical means it can't exist.

 

[Karl G.]

It ensures that it 'can', not that it 'must' in any particular instance.

But since it CAN'T, it isn't omnipotent.

 

[JoeDawg]

But since it CAN'T, it isn't omnipotent.

It can when it wants to. And yes, its a word game.

But just because our understanding of something seems paradoxical, doesn't mean that something doesn't exist.

 

[DaveC426913]

Alternately:

It can when it wants to. And yes, its a word game.

But just because our understanding of something seems paradoxical, doesn't mean that something does exist.

 

[JoeDawg]

Alternately:

All hail the Big Rock.

 

[confinement]

The fact that 'omnipotent' is paradoxical means it can't exist.

Liars are paradoxical, as I am sure you know, but they do exist.

 

[JoeDawg]

Liars are paradoxical, as I am sure you know, but they do exist.

Maybe its Schroedinger's rock.

 

[Karl G.]

Liars are paradoxical, as I am sure you know, but they do exist.

Liars aren't paradoxical. People who admit that they are liars are paradoxical.

 

[TVP45]

Suppose a right-wing radio commentator were to say, "I always tell the truth." That's not paradoxical; it's merely that we ordinary mortals are incapable of understanding the meanings of superior beings. So might it be with gods.

 

[jambaugh]

This is actually a form of Russell's paradox. (Define the set of all sets which do not contain themselves as elements and ask if that set contains itself.) You can form this when you allow both open ended definitions and circular or self reference.

This is the problem with open ended definitions e.g. omnipotence. To solve this one you just have to acknowledge that "a rock that God couldn't lift" is an ill defined concept as given any rock that could exist this God could both create and lift a larger one. Omnipotence only covers the ability to effect well defined states of reality.

 

[alxm]

It's called the http://en.wikipedia.org/wiki/Omnipotence_paradox" .

As far as Christianity is concerned, a fairly authoritative answer was given by http://www.ccel.org/ccel/aquinas/summa.FP_Q25_A3.html" in Summa Theologica's first book.

Not really an interesting question IMHO.

 

[DaveC426913]

This is actually a form of Russell's paradox. (Define the set of all sets which do not contain themselves as elements and ask if that set contains itself.)

Which is actually a subset of http://xkcd.com/468/" .

 

[confinement]

Liars aren't paradoxical. People who admit that they are liars are paradoxical.

Omnipotent beings are then not impossible in general, only those omnipotent beings that make rocks so heavy that they cannot lift them are paradoxical. In other words, any liar and any omnipotent being has the potential to be paradoxical.

 

[Helios]

What if the god moved the universe so the rock will have, in effect, been moved?

 

[Jacob Perry]

Actually if omnipotence means to be able to do anything, that means an omnipotent being could be something and not be something at the same exact time. It could do everything it couldn't do and it could be paradoxical and perfectly fit within human logic and reason. It could have no beginning and no end while having both and could and couldn't be grasped. So really, this 'paradox' proves the oneness of everything and nothing.

 

[TheStatutoryApe]

Doesn't the question rather unnecessarily anthropomorphize the omnipotent being? It seems rather pointless. Compared to a guppy I'm a rather powerful being though it may wonder what is so special about a being unable to breathe water.

 

[Count Iblis]

Suppose I have a computer powerful enough to simulate a planet like the Earth complete with people living on it down the the molecular scale. Then, I could play God in that world by modifying the simulation. I put on my virtual reality helmet, and the simulation simulates me (God) in that world.

Then, since I'm not bound by any laws of physics that operate in the virtual world, I cannot make a rock that I can't lift. But it could be that there are some actions that would cause the computer to crash when performed by me.

 

[humanino]

Doesn't the question rather unnecessarily anthropomorphize the omnipotent being? It seems rather pointless. Compared to a guppy I'm a rather powerful being though it may wonder what is so special about a being unable to breathe water.

Thank you for this comment. These is another aspect that I especially dislike in this paradox. It is restricted to an omnipotent being undergoing time (creating an object and then facing a paradox). Now consider a Leibnizian conception of omnipotent being, say for definiteness "a theory of everything" (I'm simplifying of course, but hopefully not oversimplifying). Then I hope you can see how such a conception does not undergo time. I am not defending any conception over another, neither do I think that would be interesting, I am merely trying to point out a specific example, which I think respects the general criterion provided by

Omnipotence only covers the ability to effect well defined states of reality.

By the way,

Which is actually a subset of http://xkcd.com/468/" .

was quite funny :

 

[Russell Berty]

Let Z be an omnipotent being.

Z has a rock fetish. Z thought about the large rock idea. (We are not certain if it ever happened.) Z then decided to make a rock, to be called "little mu", that was so incredibly difficult to detect that even the all-powerful Z could not detect it. Can Z ever be sure that little mu has been created?

Some people in this discussion have come close to my stance on these "paradoxes". We are arguing using our human logic. But, how can we be sure that such logic applies to the realities and lives of the all-powerful, all-knowing? What we see as a paradox might not be in some ultra-nonhuman-impossible for us to follow-hyperlogic.

Here is another lil' diddy ---
Anyone who has seen the Incompleteness Theorem may have run across this "paradox": Can Z know all the true statements of Number Theory? (If Number Theory is consistent then there is a contradiction for Z would be a truth predicate.)

My response is this... Yes because Number Theory is inconsistent, so every statement is provably true (and at the same time false.) But Z does not want us to be afraid, and if 1+1=2 leads to a contradiction we would be. So, Z makes sure that any inconsistency is out of reach. Whenever we get close to a proof of a contradiction, Z simply redoes the (faulty) Model of Number Theory so that the contradiction is further away.

 

[Hurkyl]

Here is another lil' diddy ---
Anyone who has seen the Incompleteness Theorem may have run across this "paradox": Can Z know all the true statements of Number Theory? (If Number Theory is consistent then there is a contradiction for Z would be a truth predicate.)

Gödel's incompleteness theorem doesn't say that there are no complete sets of axioms for number theory -- it merely says that such things cannot be recursively enumerable. Your "truth predicate" cannot be computed via Turing machine, and therefore there is no contradiction.

It's an easy exercise to use the axiom of choice to prove that there exist a complete sets of axioms for number theory, one doesn't need to invoke any hypothetical omnipotent/omniescient beings at all.

 

[Dadface]

Let Z be an omnipotent being.


Some people in this discussion have come close to my stance on these "paradoxes". We are arguing using our human logic. But, how can we be sure that such logic applies to the realities and lives of the all-powerful, all-knowing? What we see as a paradox might not be in some ultra-nonhuman-impossible for us to follow-hyperlogic.

.

Agreed.An omnipotent being does not have to conform to our logic and understanding.It is the definition of the word itself that raises the apparent paradox.

 

[Russell Berty]

Gödel's incompleteness theorem doesn't say that there are no complete sets of axioms for number theory -- it merely says that such things cannot be recursively enumerable. Your "truth predicate" cannot be computed via Turing machine, and therefore there is no contradiction.

It's an easy exercise to use the axiom of choice to prove that there exist a complete sets of axioms for number theory, one doesn't need to invoke any hypothetical omnipotent/omniescient beings at all.

I agree. Perhaps I was a bit sparse on my description. The real "paradox" comes in when we ask "Can Z construct a formula in arithmetic that is a truth predicate?" For, surely, Z can know all true statements, and being all-powerful, Z can make arithmetic statements that say define whatever subsets of N Z desires.

 

[Moridin]

Yes, the concept of omnipotence as defined in the opening post leads to absurd contradictions, and therefore, cannot stand.


Russell Berty,

Some people in this discussion have come close to my stance on these "paradoxes". We are arguing using our human logic. But, how can we be sure that such logic applies to the realities and lives of the all-powerful, all-knowing? What we see as a paradox might not be in some ultra-nonhuman-impossible for us to follow-hyperlogic.

There is no "human" logic. There is only logic. Horses do not have any different, equally valid, method of coherent argumentation. To say that an entity is "above" logic makes the argument fall apart at once -- if it is above logic, then it can change logic arbitrarily and make it invalid.

 

[DaveC426913]

There is no "human" logic. There is only logic. Horses do not have any different, equally valid, method of coherent argumentation. To say that an entity is "above" logic makes the argument fall apart at once -- if it is above logic, then it can change logic arbitrarily and make it invalid.

Yeah, I think his "human logic" versus "hyperlogic" is really better defined as "human-conceivable premises" versus "omniscient-cognizant premises".

If we humans had access to the knowledge (premises) of the omniscient-dude, our logic would more closely fall in line. But our premises are flawed.

 

[Hurkyl]

My response is this... Yes because Number Theory is inconsistent, so every statement is provably true (and at the same time false.) But Z does not want us to be afraid, and if 1+1=2 leads to a contradiction we would be. So, Z makes sure that any inconsistency is out of reach. Whenever we get close to a proof of a contradiction, Z simply redoes the (faulty) Model of Number Theory so that the contradiction is further away.

Incidentally, it strikes me that what you are describing sounds very much like the things you can do with plain, ordinary "human logic".

 

[Russell Berty]

Incidentally, it strikes me that what you are describing sounds very much like the things you can do with plain, ordinary "human logic".

Humans have at times had to rethink their definitions when contradictions arose. But, what we can not do is restructure the actual truths or laws of reality (if there are any.) We assume that a structure we call Number Theory exists (the one that would satisfy 2nd order arithmetic) but we cannot prove it exists. It is a supposed structure of reality that we try to analyze using our "human logic". What I was trying to say is that Z can do what we cannot do. That is, Z can restructure this innate property of reality called Number Theory so that we will not run into a contradiction in are pursuits.

 

[Focus]

Humans have at times had to rethink their definitions when contradictions arose. But, what we can not do is restructure the actual truths or laws of reality (if there are any.) We assume that a structure we call Number Theory exists (the one that would satisfy 2nd order arithmetic) but we cannot prove it exists...

We cannot prove that anything exists other than yourself, what is your point?

There are no laws of reality. There is no way that we can claim 100% that reality follows X or Y. The whole of science is built on the assumption of history repeating it self.

Some people in this discussion have come close to my stance on these "paradoxes". We are arguing using our human logic. But, how can we be sure that such logic applies to the realities and lives of the all-powerful, all-knowing? What we see as a paradox might not be in some ultra-nonhuman-impossible for us to follow-hyperlogic.

This is pretty weak claim. You can argue this way with everything. Do you claim that laws of nature do not apply to a being (or maybe more)? If your answer was yes, then you need to prove that God exists. Otherwise I can make the same statement you made to claim that trees are both trees and not trees (they have tree logic).

 

[Russell Berty]

I am not sure what you are getting at in your reply.

You can prove that you exist? Then you are more clever than I, for I cannot prove that I exist.

There are no laws of reality? You can prove this?

For me, I assume that:
1) I exist.
2) Reality exists and has order; it is predictable. i.e. there are laws.

From these assumptions I deduce my theories about reality.

As far as whether or not gods exist, I cannot say. What I am trying to say is that my arguments (or human arguments) are not valid in the realm of the "all-powerful" beings (if there are any.) To argue for or against the existence of such beings is futile. For example, if the all powerful Z decides to make modus ponens invalid then where does that leave us?

 

[JoeDawg]

You can prove that you exist? Then you are more clever than I, for I cannot prove that I exist.

Don't worry, Descartes, a very clever man, already did that: I think therefore I am.

 

[Russell Berty]

Don't worry, Descartes, a very clever man, already did that: I think therefore I am.

That of course requires proving that "All things that think, exist." Which he did not prove.

 

[Russell Berty]

...Or at the very least, he would have to prove "If I think, then I exist." As well as prove that the argument form "P and P -> Q therefore Q" in valid. So, basically prove that modus ponens is valid, and he could not use modus ponens to do this (circular reasoning.) So, if he was thoroughly skeptical and considered some all-powerful demon could deceive him, then he would have to consider "What if the demon makes me believe that modus ponens is valid, or makes me believe that my arguments are valid, when my arguments are actually not valid?" Oh, not to mention he needs to define what is meant by "I" and "think" and "exist". Very tricky.

His works are impressive, no doubt, but there is much that has been learned since his time (in part thanks to him) about logic and the notion of a proof. "I think therefore I am" is not a proof that "I exist". Not without making many unproven assumptions.

 

[Hurkyl]

We assume that a structure we call Number Theory exists (the one that would satisfy 2nd order arithmetic) but we cannot prove it exists.

Of course we can. Here's a (trivial) example of such a proof:

Hypothesis: number theory exists.
From the hypothesis, number theory exists. Therefore number theory exists. QED.

(Incidentally, there are also set theoretic universes in which there is no number theory; e.g. the universe of finite sets in any model of ZFC)

That is, Z can restructure this innate property of reality called Number Theory so that we will not run into a contradiction in are pursuits.

Number theory an innate property of reality? Anyways...

Given your hypothesis, there isn't a contradiction in number theory.

If you'll accept that, in principle, we can iterate through all possible formal proofs, then it's easy to demonstrate: the fact Z can ensure none of them are contradictory proves that no contradictions can be derived, and therefore number theory is consistent.

If you reject that, in principle, we can iterate through all possible formal proofs, then the situation can be modeled with internal set theory (related to non-standard analysis), letting everything we can iterate through be the "standard" things. Then, we invoke the transfer principle: "all proofs in number theory have noncontradictory results" if and only if "all 'standard' proofs in number theory have noncontradictory results".

 

[Borek]

It occurred to me that whenever someone asks a physics question where conclusions seems to be paradoxical, it is blamed on the reference frame. I wonder if this approach can be used for the big rock as well

 

[Russell Berty]

Hypothesis: number theory exists.
From the hypothesis, number theory exists. Therefore number theory exists. QED.

You know what I meant. There is no proof that number theory exists without assuming this as a premise.

As far as iterating ALL possible formal proofs, we cannot in a finite amount of time. And as I was implying, "as we get close to a contradiction" (while enumerating proofs), Z changes the game on us.

 

[Hurkyl]

You know what I meant.

Yes I did -- and I also know you're making something out of nothing. Some premises prove the existence of number theory. Some don't. There is nothing deep on here.

As far as iterating ALL possible formal proofs, we cannot in a finite amount of time.

Then you are in the situation of my next paragraph.

 

[Russell Berty]

The point is, if you claim to have a proof of number theory, then that proof is based ultimately upon some assumptions - assumptions that are not proven, merely assumptions. So, what I am getting at is there is no "absolute" proof of number theory. It is possible that number theory is inconsistent. That is why I said, "We assume that a structure we call number theory exists but we cannot prove it exists." Some of us assume even more, such as ZFC is consistent, but it is still only an assumption.

 

[JoeDawg]

...Or at the very least, he would have to prove "If I think, then I exist."

That is true by definition and logic. To be something that thinks, one must be 'something' and therefore exist as something.

So, if he was thoroughly skeptical and considered some all-powerful demon could deceive him, then he would have to consider "What if the demon makes me believe that modus ponens is valid, or makes me believe that my arguments are valid, when my arguments are actually not valid?"

A demon *could* make you believe anything.
But its nonsensical, ie not logical, to say that a demon can do anything to you, if you don't exist. If you are nothing, then there is nothing for the demon to do.

Oh, not to mention he needs to define what is meant by "I" and "think" and "exist".

He doesn't need to do anything of the sort. Those are different questions.

Descartes wasn't dealing with the question of what existence is, or what the self is, or what thought is. It was a more basic question.
What can I know for certain?
It doesn't matter what 'I' is, or what existence is. The answer is, yes, I can know I exist. I can say I exist, because if I didn't exist, i couldn't say or think anything.

This is what is called a self-evident truth.

 

[Focus]

The point is, if you claim to have a proof of number theory, then that proof is based ultimately upon some assumptions - assumptions that are not proven, merely assumptions. So, what I am getting at is there is no "absolute" proof of number theory. It is possible that number theory is inconsistent. That is why I said, "We assume that a structure we call number theory exists but we cannot prove it exists." Some of us assume even more, such as ZFC is consistent, but it is still only an assumption.

Seeing as you keep bringing up existence about number theory, care to define how a mathematical field can exist?

No one assumes that ZFC or number theory are consistent. What do you mean by a proof of number theory? Do you mean a theorem in number theory? The proofs in number theory are absolute, in the sense that if you accept the axioms then you must accept the theorem. You can say the same about the statements such as, if evolution happened then evolution happened which is again absolute. They are all tautological statements.

 

[Russell Berty]

Before I attempt a reply, I will give a little quiz. Defend your answers.


1) Does the number 1 exist?

2) Is it true that 1 + 1 = 2?

3) Is it possible that 1 + 1 = 2 is false?

4) Can you doubt that 1 + 1 = 2?

5) Can you doubt that 1 exists?

6) Can you prove that a statement is true without knowing the meaning of the statement? [For example, Prove that "XqXX$amNNmM yyty3wjj: kXzQQp"]

7) If a statement cannot be proven true, does that mean that the statement is false?

8) Can you prove the following statement? "There is no proof of this statement."

9) Is it possible that number theory is inconsistent (contains contradictions)?

10) Is the mind capable of making a mistake?

11) Is it possible to doubt a statement is true even though the mind claims that statement is true?


I will mention that many philopshers since Descartes have argued over the validity of Cogito Ergo Sum. To the present day there is no agreement on the validity. For a start, read Kant, Hume, and Russell.

On Number Theory: I was talking about a proof that Number Theory is consistent when saying "proof of Number Theory." The point I was making was that we do not know whether it is consistent or not (without making assumptions such as "it is consistent".) By existence of Number Theory, I mean "the existence of a model for Number Theory" (or you can take it further to mean existence in Plato's Universe, but I was not trying to go there.) By "model for Number Theory" I mean the "model" for a theory as defined in first order logic (see any text on first-order logic.)

 

[Russell Berty]

Since the quiz above is off the main topic, I will put it in a new post "A Little Quiz"

 

[JoeDawg]

For a start, read Kant, Hume, and Russell.

I have.
Now please, show us that you have.
What are the arguments?

 

[Russell Berty]

It is very difficult to debate the truth without having definitions for the words in "I think therefore I am."

There are many thought experiments you can come up with to bring the validity into doubt, try it. Descartes' Demon is very clever, more clever even than Descartes.

But I will try an approach that will not rely on whatever definitions are being used.


We are discussing Absolute Certainty concerning that truth of the statement "I exist." Cogito Ergo Sum is supposedly a complete validation of the statement "I exist." There is no uncertainty whatsoever in the argument or the conclusion.

I am trying to refute this.
Sorry it is a bit long, but enjoy. :)

We are in a dark alley (why? None of your business, stop asking questions!) and someone gives us a list of instructions.

They tell us "It is an algorithm that will verify any statement that is 100% certainly True." We run home to try it out.

It is called the Certainty of Truth Validator (or make up your own name) and we abbreviate it as CTV.
We are told that given a proposition p, when we run CTV on p, denoted CTV(p), either CTV will halt and say 100%T, or it will halt and it will say <100%T (not absolutely certainly true), or it will never halt.
We are guaranteed that if p is 100%T then CTV(p) will halt and we will get the response 100%T.

So, there I am with the CTV. I wonder if CTV is itself reliable 100%. I want to make sure that CTV will never say 100%T when the input proposition is not 100%T. So I try the following:

CTV( It is not the case that, if CTV(p) outputs 100%T then p is not 100%T )

I get 100%T. Not surprising, look who I am asking.

I then grow a little concerned. I am the one who just ran CTV! Am I reliable? I will ask CTV.

CTV( For all propositions p, I run CTV(p) without error )

I run the algorithm, holding my breath. Still holding...
Maybe it stops and says 100%T (phew) or maybe it does not (uh-oh.)

But, even if it says 100%T, how do I know I ran THAT input correctly?

For brevity, let A0 := "For all propositions p, I run CTV(p) without error" I just ran CTV(A0).

So, I run

CTV (I run CTV(A0) without error).

Again I hold my breath... Even if I get 100%T, did I run that correctly?

Recursively define the following propositions:
A(n+1) = "I run CTV(An) wihtout error" {for n = 0, 1, 2 ...}

I just ran CTV ( A1 ).
But, still uncertain of MY OWN ABILITY to run CTV without error, I continue.

CTV ( A2 ), CTV ( A3 ), ...

Regardles of the results (I am being nice to myself saying that CTV says I run it without error in these cases) I will forever be UNCERTAIN OF MY OWN ABILITY TO RUN CTV.

To be fair, I was asking alot, for the initial satement said "I run CTV(p) without error on ALL propositions p".

So, let us try something more specific.

I run
CTV (I exist).

Now, to cover whatever arguments people claim show "I exist" is 100%T, I will allow the instructions for CTV to contain that list for just this situation. (i.e. see CTV Manual pg. 4,329 line 78 "When the input is "I exist" follow these...)

OK. So I run CTV (I exist) [Define B0 := "I exist" ]

I get back 100%T (otherwise...omg)

But, did I run it correctly! So I run CTV (I run CTV (I exist) without error). I get 100%T. Goody.
But I still might have made a mistake, I am not ABSOLUTLEY CERTAIN.

Define B(n+1) = "I run CTV(Bn) without error" {for n = 0, 1, 2 ...} I ran CTV(B1). So now I run
CTV(B2), CTV(B3), ...

always wondering whether I made a mistake somewhere; always just a little uncertain; never absolutely certain that the statement "I exist" is true.

 

[JoeDawg]

always wondering whether I made a mistake somewhere.

I think you're getting hung up on language.

'Cogito ergo sum' really has no 'I' in it.

"Thinking exists" is a perfectly good translation.

Even, 'doubting exists', if one wants to use Descartes evil demon example.

And if thinking exists, then thinking is a thing.
We can call that thing 'I', we can call it x, or Fred, or any other word.

You don't have to know the nature of self, you don't have to know the nature of thinking, nor do you need to define the nature of existence.

Its more basic than that. Thinking exists is the same as saying thinking occurs, or thinking happens. And this is unavoidable... its a self-evident truth.

If you doubt thinking happens, you are thinking and doubting. So its self-contradicting.

 

[Russell Berty]

For more on the issues with claiming "absolute truth" of anything, even that "thinking exists" or "I exist", see the following.

1933, Idealistic logic
a study of its aim, method, and achievement
by C. R. Morris

It can found online.


http://books.google.com/books?id=ED...eF8csP&sa=X&oi=book_result&ct=result&resnum=2