Gödel's theorem doesn't have anything to do with Einstein's theories or any other theory. The reason is that Gödel's theorem is relevant about whether statements in a particular logical system can be proved to be decidable on the basis of the axioms of the system. The statements made within any physical theory are of different nature: it is known that the theoretical descriptions are incomplete and the question is how to describe the degrees of freedom that complete them, at any given level of description. On the basis of what is, already, known and measured, it is possible to deduce, whether new effects could appear in any gven context-if they haven't been detected in other contexts-or are such deviations rather due to the experimental procedure itself.
The two statements presented are, unfortunately, incorrect and the correct statements can be found in standard textbooks and courses on the subject.
Gödel's theorem doesn't have anything to do with Einstein's theories or any other theory. The reason is that Gödel's theorem is relevant about whether statements in a particular logical system can be proved to be decidable on the basis of the axioms of the system. The statements made within any physical theory are of different nature: it is known that the theoretical descriptions are incomplete and the question is how to describe the degrees of freedom that complete them, at any given level of description. On the basis of what is, already, known and measured, it is possible to deduce, whether new effects could appear in any gven context-if they haven't been detected in other contexts-or are such deviations rather due to the experimental procedure itself.
The two statements presented are, unfortunately, incorrect and the correct statements can be found in standard textbooks and courses on the subject.
It is unclear to me if you are talking of the incompleteness of the special theory of relativity or of the general theory.
I don't think the fact that in (special) relativity there is no "perfect" rigid body, is a drawback of the theory of relativity. In any case the rigid body is an idealization. Physical rigid bodies are not really rigid, nor they are continuous...they are made of atoms. The fact that the there are no rigid bodies in relativity can be viewed as a refinement of our understanding of nature.
The Ehrenfest paradox seems to be an idea that floated around the time Einstein was thinking of the equivalence principle, and maybe it was one of the factors that brought Einstein to formulate the general theory of relativity. The Ehrenfest paradox can be considered a limit of special relativity, since special relativity deals with inertial fames of reference... and an observer on the rim of the disk would be in a non inertial frame. There are some authors that tried to explain Ehrenfest paradox in the realm of special relativity but I don't know if it was successful.
As for the general theory of relativity is it incomplete? Maybe. It certainly seems to have drawbacks. From what I understand it is difficult to reconcile it with Quantum Mechanics, and one could argue that there is no satisfactory way to reconcile General Relativity with Quantum mechanics.
A theorem (if any) that can describe or take us to the ultimate reality (or ubiquitous Consciousness) will be the complete one. All else, ion my opinion, should be considered incomplete.
Manuele
Certainly it is difficult to reconcile it (GR) with Quantum Mechanics, but don't argue that there is no satisfactory way to reconcile General Relativity with Quantum mechanics. A lot of clever people are working on this, don't be so pessimistic!
Matts,
I don't think I am being pessimistic. I am being optimistic! If there was nothing left to understand, that would be the end of physics. A world with nothing left to understand would be a really sad place.
There are certainly many very smart people working on reconciling GR and QM. I have to claim ignorance on this subject. I do know, from the outside, some people working in String Theory and loop quantum gravity, but are these satisfactory theories? Do they reconcile QM and GR? I don't know enough to answer these questions. But, maybe you can.
Manuele
I'll give you an explanation but not a solution because none is known yet.
The Einstein-Hilbert action (from which the Einstein equation is derived) is not renormalizable, therefore standard general relativity cannot be properly quantized. However, renormalizability can be cured by adding higher order terms in curvature invariants. This is similar to the situation in quantum field theory where renormalization is some procedure to remove infinities in calculations. If the Lagrangian contains combinations of field operators of high enough dimension in energy units, the counterterms required to cancel all divergences proliferate to infinite number, and, at first glance, the theory would seem to gain an infinite number of free parameters and therefore lose all predictive power, becoming scientifically worthless. Such theories are called nonrenormalizable.
The Standard Model of particle physics contains only renormalizable operators, but the interactions of general relativity become nonrenormalizable operators if one attempts to construct a field theory of quantum gravity in the most straightforward manner, suggesting that perturbation theory is useless in application to quantum gravity.
The endeavor to remedy the renormalizability of the Einstein-Hilbert action implies adding functions of the Ricci scalar R which become important at late times and for small values of the curvature. This is required in order to avoid conflicts with constraints from the solar system or the Galaxy..
About rigid bodies in G.R. There are solutions of Einstein's equations for stationary mass distributions, but that begs the question. Here is a better wayto throw some light on the question. Consider a galaxy that rotates about an axis, in the manner of a rigid body. Wer are now in a position to study such systems. How does this system cope with the high velocities at the perifery; how does it avoid motions that are faster than that of light? This is one my projects at the moment.
Dmitri,
About your opening sentence, that Gödel's theorems somehow imply incompleteness of Einstein's theory.
My point is, and that is along the lines of Stam's post, that you are comparing two entirely different notions of 'completeness'.
There is the idea of completeness in mathematics, roughly meaning that every true formula of an axiomatic system can be proven from its axioms. This is the notion of completeness in Gödels theorems. It has nothing to do with the physical world.
And there is the notion of completeness in physics, meaning that (i) every element of reality, predicted with certainty by a theory, indeed exists, and (ii) vice versa, every element of reality must have a counterpart in the theory. This is defined in the EPR paper.
So Gödels theorems on mathematical (in)completeness of certain axiomatic systems have no implications for the physical (in)completeness of Einstein's relativity (special or general).
Christian Fronsdal,
In reply to your comment and looking at your recent preprints, it appears to me that you are simplifying the dynamics of spiral galaxies. The Universal Rotation Curve found by P. Salucci and collaborators indeed shows that dark matter has some interaction with baryonic matter, or perhaps it is exhibiting velocity-dependent self-interactions. To solve this problem for spirals, a condition is that the solution must not conflict with observations of galaxy clusters. Even so, observations of dwarf galaxies are not well understood.
Marcoen,
Godel's theory does imply that Einsteins theory is incomplete. If you think other wise then there is one glaring example "the theory breaks down" in what we call a black hole. At it limits it does not work.
This means to any reasonable person that the theory is incomplete. If it were totally complete it would describe everything everywhere and not break down at points along the way. Math has its limits. Math does not describe nature it only describes how we calculate within the limits of the math system the predictive qualities of our understanding. It is a great approximation at low pressures and or low volumes but it is not the end all. If it were it would have been the Unified Field Theory that Albert Einstein spent the last several decades of his life looking for.
Do not discredit Albert Einsteins last several decades of work as rubbish. If he himself had not thought the theory of Relativity was "inexact" he would have spent that time working on something else. The most brilliant man of the last century is mostly ignored after he realized his own theory was limited. Lets not ignore the signes of the time and the work that has been showing the cracks in the old foundations of our science. Are we to be the naysayers of this century and tell the Albert Einsteins of our time that they are wrong? I am not saying this is exactly correct but what I am saying is that the ideas here have too much merit to just let go or to just say your wrong.
George,
from your reply I see that you haven't fully understood what it means that there are two different concepts of completeness involved here. Gödels theorem - not theory - has no implication for the (in)completeness of Einstein's relativity: that's fact, not opinion. The example that you give, even if it were true, is by no means a proof to the contrary.
Marcoen,
I understand your concern but I have to disagree with you.
If we look at the logic of what you just said for a minute.
Einstein"s "theory" is a mathematical model of how the universe has to work given the understanding that we think we have of time, space, and gravity. This is how we describe the universe in math. Godel pointed out in his "theorem" (Please read below for the definition) that all models that are mathematical in nature and have limits, postulates, and rules are by definition "incomplete" This goes to point out that no system using these conditions could therefore prove everything. Any system with limits (hence divide by zero) or anything else is limited. Please tell me what I am missing?
There is a lack of logic used today in science that I do not understand. Logic tells us that if two theories (General Relativity, and Quantum Mechanics) Both the most proven theories of all time do not match when it comes to the math then there is something wrong with either one of them or both of them and this is the proof that Godel was correct.
The first logic's class that I took in High School showed me this. If two things are not the same they can not be equal. One of them or both of them is wrong. They are both great approximations at low pressures and densities but they are not correct on the largest of scales therefore they are not great predictors of what the things on the largest of scales look like.
DEFINITION:
theorem |ˈθēərəm; ˈθi(ə)r-|
noun Physics & Mathematics
a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths.
• a rule in algebra or other branches of mathematics expressed by symbols or formulae.
The main difference between a theorem and theory is that the later is never proven or it is no longer a theory.
George,
Since you asked me to tell you what you are missing, my reply is this: what you are missing is the necessary background in the foundations of mathematics and the foundations of physics. A High School class in logic doesn't cut it. Your paraphrase of Gödels theorem ("Gödel pointed out ...") is incorrect, and so is the conclusion that you draw from there ("This goes to ..."). I'm sorry but I cannot put it any better. Unfortunately, I do not have the time to give a detailed treatise on this material here and now. But a major point is that Gödels theorem is about formal proofs in axiomatic systems of the form 'A1, ..., An |- T', it has nothing to do with proving things about the physical world. I'll have to leave it at this. But please note that I appreciate your interest in this matter!
Gödel's theorem is completely irrelevant for physics.
Then, if there is a perfect rigid body or not is completely irrelevant for the validity of a physical theory too.
And in GR there are no inertial systems, Trying to construct some contradiction in GR based on notions of SR makes no sense.
@George, you write "The first logic's class that I took in High School showed me this. If two things are not the same they can not be equal. One of them or both of them is wrong."
1+1=2 and 1+2=3 are different equations, not the same.
"Logic tells us that if two theories (General Relativity, and Quantum Mechanics) Both the most proven theories of all time do not match when it comes to the math then there is something wrong with either one of them or both of them and this is the proof that Godel was correct."
Yep, there is something wrong with one of the two theories or above. But this would in no way prove Gödels theorem - it is completely unrelated, because it is not about correctness, but about completeness, and could be, at best, only an example, not a proof. And, moreover, the theorem does no longer need any proof, it has already been proven by Gödel.
Marcoen,
I have more than a high school class in logic and math. I do have a masters degree yet you are correct I do not have a pH D in math nor do I want one. The conceptual ideas that are put forward by Godel are valid in all works universally. You should not choose to ignore these ideas as only part of a mathematical base of knowledge.
I work in Quantum Chemistry and have seen many different results from what I have been told is possible. This has lead me to some good understanding of how much we do not know in the world of sub-atomic quantum physics. All though most of this is not relevant to relativity it is relevant to Godel and incompleteness.
Think as a philosopher would on this issue. Godel is not just a mathematician but was a great philosopher. As scientist if we choose to think that we know it all as correct, when it is not, then we set ourselves up to fail
My work is leading me to a better model of the atom and much more but that is not what this talk is about.
I still respect your opinion but think you should look at the broader conceptual picture behind the ideas.
George,
I was orginally educated in physical (quantum) chemistry, too. But the foundations of physics really is an entirely different ball game. Only now, after some 18 years in the field and after extensive re-education in mathematics, physics, and philosophy, I dare to say that I have enough background to work as a professional in the field.
That aside, looking from a broader perspective to the present issue, as you suggest, isn't always helpful, in particular not in the present case. Let me illustrate that with an example. As a chemist you undoubtly know that HDPE is a strong thermoplastic. Now suppose that layman William says: so if we put that in drug tablets, we get a strong medicine. But you would then answer: absolutely not, the one concept 'strong' has nothing to do with the other concept 'strong'. But William would then say: I disagree, you have to keep an open mind on things, etc. The same here in this discussion: the talk involves two different concepts of completeness. Both are established concepts, but the one has nothing to do with the other. Open-mindedness doesn't change that.
Einstein's theory seems to be complete until one includes other non-scalar fields. Lagrangian of the combination of gravitational and spinor fields is not self-consistent. The only way out is to add one more term which is Riemann-Riemann convert, that everything is OK, but I have no idea how to solve the corresponding modernization of the Einstein equation. Nor I see what can change of it because such an addition is exactly the integrand of the Gauss-Bonnet-Chern theorem.
Kurt Gödel's theorem asserts no formal system can be complete and consistent at the same time. Lobachevsky developed non euclidean geometries assuming different statements for the axiom of the parallels. All this geometries are consistent and they are valid in their own realms where their axiom set holds true. If we look at physical theories as formal systems, these are based not in axioms but in principles which have been tested for validity in experiments, this sets a fundamental difference between logical formal systems and physical theories. It is nature who dictates the validity of a principle. Unlike in logical formal systems if a physical theory is based on a principle no other theory can be based on this principle negation because nature would say it's not true.
A theory is never reality but a model to help us understand nature, irrespective of any Gödel's conclusion.
Juan,
The answer you give is true. The problem with natural systems is they always have limits. Just like we see limits in the physical laws. You have to say then that any natural system would be in accord to Godel's main points. That any system with limits and definitions is by definition incomplete.
George E.
Any physical theory would be incomplete I agree. In logic and maths you can find undecidable statements, adding one of this statement's assertion or negation to a system as a new axiom would yield different systems logically consistent and incomplete. I would stress that any physical proposition about nature (principle) is decidable if it were not it wouldn't pertain to the domain of physics, it is nature who decides if a physical statement is true or not.
Juan,
I think that you are correct. The only true system is a natural system. All the man made systems are incomplete.
Nathan,
I think that the problem we have with what we call dark matter is that we are not looking right under our noses. If we were to look at the properties of matter and how they are affected by gravity with the factors of wavelength in them then we may see some types of matter that are not really exotic but rather plane. If we drop matter to its lowest energy state and we look at even the Bose Einstein Condensate as an example, you see that gravity has a larger affect on matter as its energy state goes to zero. This means that if the dark matter is just regular matter with a near zero energy state (tired matter) it would actually have a higher gravitational potential than matter that is in motion or with higher energy states.
The dark matter could just be described as tired matter. This would explain why we do not see it but we feel its presents. If it is not radiating we would not see it as standard matter.
Lets say that if the radiation was less than the 2.745 degrees above zero then we would have trouble seeing it as it would be swamped by the back ground.
Nathan,
String theory can explain almost anything. If I add a few more dimensions to it you can explain the existence of magic as well. That is the problem with string theory. It is not reality, as reality has 3 spacial and one time dimension and theories that do not take from that can be made to explain almost anything.
If I add enough component to even a line in space I can explain a plain and if I add more to it I can explain a sphere. My point is that the theory is only useful in the abstract and not in reality. Sure I can explain lots but I want reality.
Our understanding of the physical world is great down to an atomic level but when we try to get to the sub-atomic we are just guessing and it is not always a good guess.
Nathan,
It will take young people like you to over though the old guard in Physics before we get real change. The old farts in the field have built to much of their PHD's on the way things use to be to let go even when it is wrong.
My favorite example of this is the fact that General Relativity and Quantum Mechanics (the most proven of all theories in Physics) do not match each other and when we try we get things like divide by Zero and undefined. If we can not see that the classical and quantum have to match before the theories are complete then we are not willing to give up to reality yet. It is a logical imperative and yet no one will move from either side to resolve the issue.
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