# Theory of everything vs Godel's incompleteness theorem.

• Jim Kata
In summary, the conversation discusses the implications of Godel's incompleteness theorem on the possibility of a theory of everything. While some argue that the theorem suggests a TOE could never exist, others point out that it only applies to mathematical systems and a physical TOE may not need to satisfy its assumptions. The conversation also touches on the idea of self-referential statements and the need for a sufficiently rich formal system in constructing a TOE.
Jim Kata
Wouldn't Godel's incompleteness theorem imply that we could never have a theory of everything?

Jim Kata said:
Wouldn't Godel's incompleteness theorem imply that we could never have a theory of everything?

In what way does a theory of everything satisfy the assumptions of Godel's theorem?

Stephen Hawking made some remarks on this but they were all qualitative. fzero is right, to be rigorous you would need to show it that your TOE satisfies the assumptions of Godel's theorem.

But also, that doesn't mean a TOE doesn't exist. His incompleteness theorem just shows that you can't prove it to be consistent, if I'm not mistaken.

Godel's incompleteness theorem is only relevant when you're trying to derive every mathematical statement that's true. Finding a TOE is the opposite effort. We're only trying to find one equation.

Jim Kata said:
Wouldn't Godel's incompleteness theorem imply that we could never have a theory of everything?
No - Godel's theorem is about mathematics. TOE would be a physical theory - it is unlikely the math involved would require considering Godel's theorem.

Jim Kata said:
Wouldn't Godel's incompleteness theorem imply that we could never have a theory of everything?
It depends on what exactly one means by "everything". According to the Godel's theorem, any physical theory, say about electrons, will contain a self-referring statement like

"This property of electrons cannot be proved."

which is true but cannot be proved. But is it really a problem for physics if such a physically-empty statement cannot be proved? I don't think so.

fzero said:
In what way does a theory of everything satisfy the assumptions of Godel's theorem?
A physical theory of "everything" probably contains all integer numbers, which is an assumption of the Godel's theorem.

Demystifier said:
It depends on what exactly one means by "everything". According to the Godel's theorem, any physical theory, say about electrons, will contain a self-referring statement like

"This property of electrons cannot be proved."

which is true but cannot be proved. But is it really a problem for physics if such a physically-empty statement cannot be proved? I don't think so.

This is not a self referential statement. And why is it true? The property of electrons may be true, but this proposition can be false.

martinbn said:
This is not a self referential statement.
Maybe I should rewrite the statement as

"This statement on electrons cannot be proved."

It is self-referential because the word "this" in this statement refers to this statement itself. See also
http://en.wikipedia.org/wiki/Self-reference

martinbn said:
And why is it true?
Assume the opposite, that it isn't true. Then that statement CAN be proved, because "can be proved" is the opposite of "cannot be proved". But if it can be proved (and if TOE cannot prove a false statement), then it is true. However, the result that it is true contradicts the initial assumption that it isn't true, so the initial assumption must be wrong. Therefore, the statement must be true.

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Demystifier said:
Yes it is. That's because the word "this" in the sentence

"This property of electrons cannot be proved."

refers to this sentence itself.

No, it refers to some property of electrons, not to itself, unless you are saying that this proposition itself is a property of electrons!

Assume the opposite, that it isn't true. Then that sentence CAN be proved, because "can be proved" is the opposite of "cannot be proved". But if it can be proved (and if TOE cannot prove a false statement), then it is true. However, the result that it is true contradicts the initial assumption that it isn't true, so the initial assumption must be wrong. Therefore, the sentence must be true.

Here you are mixing things up. You are confusing the 'can be proved' as referring to the the property of electrons or the proposition.

If you were right, it would mean that any formal system is incomplete, which is not true. So you must be mistaken.

Martinbn, in the meantime I have editted my post #9. Could you read it again?

martinbn said:
If you were right, it would mean that any formal system is incomplete
That, indeed, is essentially what the Godel theorem says: that any (sufficiently reach) consistent formal system is incomplete.

Of course, most people are shocked and cannot believe it is true when they hear about the Godel theorem for the first time. So it seems that you are not an exception, which is OK.

OK, you changed it to "This statement on electrons cannot be proven.", electrons are inconsequential, so you may say "This statement cannot be proven.", which is the original Goedel's statement, but you need to have enough structure of your formal system in order to have that statement. So your sentence before that "According to the Godel's theorem, any physical theory, say about electrons, will contain a self-referring statement like..." is not necessarily true.

Demystifier said:
That, indeed, is essentially what the Godel theorem says: that any (sufficiently reach) consistent formal system is incomplete.
Exactly, and you were trying to imply that the theorem applies always.
Of course, most people are shocked and cannot believe it is true when they hear about the Godel theorem for the first time. So it seems that you are not an exception, which is OK.
This is a cheap shot. I thought we are having a discussion, but if you are going to be condescending and patronizing, well then...

Martinbn, I have to withdraw some of my statements and admit that you are right. In principle, the physical TOE does not need to be sufficiently reach to satisfy the assumptions of the Godel theorem.

In fact, it is not difficult to construct a toy model for such a TOE. For example, TOE may be an algorithm which for every initial position x_0, initial velocity v_0, and time t as the inputs gives the final position x as the output, according to the rule
x = x_0+v_0 t
That's ALL what this TOE does, and there may be no need for anything else.

The Pandora box of possible problems, including those considered by the Godel theorem, opens up when you start to ask questions such as
"Can the formula above give x for ANY value of t?"
or
"Can someone prove that it can?"
For, if someone proves that for you, then you may ask similar questions
"OK, but can you prove that for ANY v_0 there is ... blah blah ...?"
If you open up that Pandora box, sooner or later you will arrive at a statement of the Godel type.

However, my point is that you do NOT need to ask such general questions. All the questions my TOE above needs to answer are of the form
"What is x for this or that particular finite values of x_0, v_0 and t?"
And as long as you stick to questions of this type and nothing else, there will be no any sign of "incompleteness" in this TOE.

Demystifier said:
However, my point is that you do NOT need to ask such general questions. All the questions my TOE above needs to answer are of the form
"What is x for this or that particular finite values of x_0, v_0 and t?"
And as long as you stick to questions of this type and nothing else, there will be no any sign of "incompleteness" in this TOE.

That is also my opinion, my remark was more pedantic than anything else.

martinbn said:
That is also my opinion, my remark was more pedantic than anything else.
I'm glad that eventually we arrived at an agreement.

If a theory of everything happens to include a sufficient amount of facts about integers that it meets the conditions of Godel's theorem, then there will indeed be truths in the language of the theory which cannot be proven by the theory, but these will be truths about the integers, not truths about the physics of the theory. So in particular, Godel's theorem will NOT lead to unprovable properties of electrons.

lugita15 said:
If a theory of everything happens to include a sufficient amount of facts about integers that it meets the conditions of Godel's theorem, then there will indeed be truths in the language of the theory which cannot be proven by the theory, but these will be truths about the integers, not truths about the physics of the theory. So in particular, Godel's theorem will NOT lead to unprovable properties of electrons.
I don't think that in a single theory one can make a clear separation between "mathematics" and "physics".

Demystifier said:
I don't think that in a single theory one can make a clear separation between "mathematics" and "physics".
But all Godel's theorem says, essentially, is that no axiomatic theory can prove all the (first-order) truths about the integers. So as such, it has absolutely no bearing on whether the theory has gaps concerning physics. Now you may be wondering how e.g. the set theory ZFC is susceptible to Godel's theorem, even though it only talks about sets, not integers. The reason is that you can represent integers in terms of sets, for instance defining zero as the set of the empty set. Thus Godel's theorem tells us that no axiomatic theory can prove all the truths about sets either.

So similarly, you can imagine some weird hypothetical TOE in which integers are represented not as themselves, which would be ho-hum physics wise, but rather in terms of he physical content of the theory. For instance, you can have an infinite number of particle types, each type corresponding to one natural number. Then various properties of numbers would correspond to various properties a particle can possess, and Godel's theorem would tell us that there is some particle property such that the proposed TOE cannot tell which particles possesses it and which don't. But this is a rather far-fetched scenario, and it would only become an issue if the theory proved enough facts about the natural numbers for Godel's theorem to apply, and depended strongly enough on all knowing all the truths of number theory, both of which are very unlikely possibilities.

## 1. What is the Theory of Everything?

The Theory of Everything is a theoretical framework that seeks to unify all the fundamental forces and particles in the universe into a single, comprehensive theory. It aims to explain and predict all physical phenomena, from the smallest subatomic particles to the largest structures in the universe.

## 2. What is Godel's Incompleteness Theorem?

Godel's Incompleteness Theorem is a mathematical theorem that states that any formal system of mathematics is either incomplete or inconsistent. This means that there will always be true statements within the system that cannot be proven, and there may also be contradictions within the system.

## 3. How do these two concepts relate to each other?

The Theory of Everything and Godel's Incompleteness Theorem are both attempts to understand and explain the fundamental laws and principles that govern the universe. However, they approach this goal from different angles - the Theory of Everything seeks to unify all physical phenomena, while Godel's Incompleteness Theorem deals with the limits of formal systems and mathematics.

## 4. Can Godel's Incompleteness Theorem disprove the Theory of Everything?

No, Godel's Incompleteness Theorem does not disprove the Theory of Everything. While it may pose limitations on our ability to fully understand and describe the universe, it does not invalidate the concept of a unified theory.

## 5. Are there any potential implications of Godel's Incompleteness Theorem on the search for a Theory of Everything?

Yes, Godel's Incompleteness Theorem has led some scientists and philosophers to question whether a Theory of Everything is even possible. It suggests that there may always be aspects of the universe that are beyond our understanding and that a complete, all-encompassing theory may be unattainable.

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