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Jim Kata
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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?
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 likeJim Kata said:Wouldn't Godel's incompleteness theorem imply that we could never have a theory of everything?
A physical theory of "everything" probably contains all integer numbers, which is an assumption of the Godel's theorem.fzero said:In what way does a theory of everything satisfy the assumptions of 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.
Maybe I should rewrite the statement asmartinbn said:This is not a self referential statement.
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.martinbn said:And why is it true?
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.
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.
That, indeed, is essentially what the Godel theorem says: that any (sufficiently reach) consistent formal system is incomplete.martinbn said:If you were right, it would mean that any formal system is incomplete
Exactly, and you were trying to imply that the theorem applies always.Demystifier said:That, indeed, is essentially what the Godel theorem says: that any (sufficiently reach) consistent formal system is incomplete.
This is a cheap shot. I thought we are having a discussion, but if you are going to be condescending and patronizing, well then...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.
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.
I'm glad that eventually we arrived at an agreement.martinbn said:That is also my opinion, my remark was more pedantic than anything else.
I don't think that in a single theory one can make a clear separation between "mathematics" and "physics".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.
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.Demystifier said:I don't think that in a single theory one can make a clear separation between "mathematics" and "physics".
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.
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.
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.
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.
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.