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.
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.
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.
A physical theory of "everything" probably contains all integer numbers, which is an assumption of the Godel's theorem.
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.
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
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. :tongue2:
No, it refers to some property of electrons, not to itself, unless you are saying that this proposition itself is a property of electrons!
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?
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.
Exactly, and you were trying to imply that the theorem applies always.
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?"
"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.
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.
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 possess 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.
Separate names with a comma.