Impact of Gödel's incompleteness theorems on a TOE

[PhysDrew]

Well I've searched this forum and didnt see this topic before, so I was wondering what the general consensus is on the impact of Godel's theorem (if any) on a possible TOE?
I'm still wading through it all so just wanted some other opinions...

 

[mathman]

Godel's theorem is a mathematics foundation subject. It has nothing to do with physics (TOE).

 

[PhysDrew]

Godel's theorem is a mathematics foundation subject. It has nothing to do with physics (TOE).

Mmmm but put simply physics is based on mathematics, it is written in the language of mathematics. Stephen Hawking has addressed the problem, and has conceeded that it does apply;

http://www.hawking.org.uk/index.php/lectures/91

So why has it nothing to do with physics?

 

[Chronos]

The philosophy forum is more appropriate for this thread.

 

[Chalnoth]

Well I've searched this forum and didnt see this topic before, so I was wondering what the general consensus is on the impact of Godel's theorem (if any) on a possible TOE?
I'm still wading through it all so just wanted some other opinions...

The basic argument that Hawking was making on this subject a few years back is that even if we do discover a theory of everything, physicists would still have a lot of work to do, because due to Goedel's incompleteness theorem, it would be impossible to ever discover all of the consequences of the theory of everything. Thus it's basically an argument that a discovery of a theory of everything, if it occurs, would still leave physicists with lots of work to do.

 

[Kevin_Axion]

I thought Godel's Incompleteness Theorem was only applicable to axiomatic systems. Does Physics have such systems or only pure mathematics?

 

[Chalnoth]

I thought Godel's Incompleteness Theorem was based applicable to axiomatic system. Does Physics have such systems or only pure mathematics?

I think the expectation is that a theory of everything, whatever that may be, would be an axiomatic system.

 

[Kevin_Axion]

What would justify it being an axiomatic system? Pure Mathematics relies on the notion of axioms, Physics requires experimentation.

 

[Chalnoth]

What would justify it being an axiomatic system? Pure Mathematics relies on the notion of axioms, Physics requires experimentation.

The point is that axiomatic systems are not arbitrary. Some sets of axioms produce well-behaved mathematical structures, some do not. So the sorts of mathematical structures available is limited in some sense.

The question, then, is which of these mathematical structures is isomorphic to reality. If we ever do manage to eliminate all but one mathematical structure as describing our reality, then that will be our theory of everything. And that one mathematical structure will probably be an axiomatic system.

 

[bcrowell]

There is a good book on this kind of thing: Franzen, Godel's Theorem: An Incomplete Guide to Its Use and Abuse. There are lots of different reasons why Godel's theorem is not relevant to the search for a TOE:

Physics is not axiomatic system.

We don't have a TOE, so we don't know whether Godel's theorem would apply to it, even assuming that it could be made into an axiomatic system. Godel's theorem only applies to certain types of axiomatic systems. For example, it does not apply to elementary Euclidean geometry, which can be proved to be consistent.

It is possible to prove that one axiomatic system is equiconsistent with another, meaning that one is self-consistent if and only if the other is. If we had a TOE, and we could make it into an axiomatic system, and it was the type of axiomatic system to which Godel's theorem applies, then it would probably be equiconsistent with some other well known system, such as some formulation of real analysis. Any doubt about the self-consistency of the TOE would then be equivalent to doubt about the self-consistency of real analysis -- but nobody believes that real analysis lacks self-consistency.

Finally, there is no good reason to care whether a TOE can't be proved to be self-consistent, because there are other worries that are far bigger. The TOE could be self-consistent, but someone could do an experiment that would prove it was wrong.

 

[atyy]

There is a good book on this kind of thing: Franzen, Godel's Theorem: An Incomplete Guide to Its Use and Abuse. There are lots of different reasons why Godel's theorem is not relevant to the search for a TOE:

Physics is not axiomatic system.

We don't have a TOE, so we don't know whether Godel's theorem would apply to it, even assuming that it could be made into an axiomatic system. Godel's theorem only applies to certain types of axiomatic systems. For example, it does not apply to elementary Euclidean geometry, which can be proved to be consistent.

It is possible to prove that one axiomatic system is equiconsistent with another, meaning that one is self-consistent if and only if the other is. If we had a TOE, and we could make it into an axiomatic system, and it was the type of axiomatic system to which Godel's theorem applies, then it would probably be equiconsistent with some other well known system, such as some formulation of real analysis. Any doubt about the self-consistency of the TOE would then be equivalent to doubt about the self-consistency of real analysis -- but nobody believes that real analysis lacks self-consistency.

Finally, there is no good reason to care whether a TOE can't be proved to be self-consistent, because there are other worries that are far bigger. The TOE could be self-consistent, but someone could do an experiment that would prove it was wrong.

But does the incompleteness theorem show that a TOE is not possible, even in principle?

 

[Kevin_Axion]

How can it show it's impossible if we don't know if a TOE is an axiomatic system? At this point it isn't really applicable.

 

[atyy]

Is a TOE not an axiomatic system?

 

[martinbn]

Even if TOE is an axiomatic system, why should it be incomplete? There are complete systems for example the Euclidean geometry.

 

[atyy]

Arithmetic is incomplete.

We probably need arithmetic in the TOE, no?

 

[bcrowell]

Is a TOE not an axiomatic system?

We don't have one yet, so we don't know. But, for example, the standard model has never been stated as an axiomatic system according to the definition used in Godel's theorems. The definition used in Godel's theorems is extremely strict. For example, here is a formal statement of Euclidean geometry: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012 Physicists never want or need to do anything with this level of formality.

-Ben

 

[D H]

PhysDrew: I take it that you are quibbling over what "everything" in "theory of everything" means. Who cares? Certainly not physicists.

All this discussion of Euclidean geometry is a bit of a red herring, for at least two reasons. #1: Even Euclidean geometry, at least where it intersections algebraic geometry, is incomplete or inconsistent. The Pythagorean theorem does it in. #2: Physics uses real numbers and more complex number systems, derivatives, all that. That places the mathematics used by physicists smack dab in the middle of a system of sufficient power so as to be subject to Godel's theorems. Once again, who cares? Physicists? No.

 

[bcrowell]

Arithmetic is incomplete.

We probably need arithmetic in the TOE, no?

It's not immediately obvious to me that we do. I know that sounds nuts, but this kind of thing is not necessarily intuitive. You would think that since the reals are a bigger, fancier mathematical system than the natural numbers, then since arithmetic is incomplete, the reals would have to be as well. But that's not the case. The elementary theory of the reals is equiconsistent with the elementary theory of Euclidean geometry, which is provably consistent. It's quite possible that a TOE could be expressed in geometrical language, without the use of any arithmetic.

 

[bcrowell]

All this discussion of Euclidean geometry is a bit of a red herring, for at least two reasons. #1: Even Euclidean geometry, at least where it intersections algebraic geometry, is incomplete or inconsistent. The Pythagorean theorem does it in.

This is incorrect. See http://en.wikipedia.org/wiki/Complete_theory for the relevant notion of completeness and incompleteness.

#2: Physics uses real numbers and more complex number systems, derivatives, all that. That places the mathematics used by physicists smack dab in the middle of a system of sufficient power so as to be subject to Godel's theorems.

This is also incorrect. See #18. The informal notion of "power" you have in mind is not the appropriate concept for discussion Godel's theorems. E.g., the reals and the complex numbers are equiconsistent, because you can model the complex numbers using the reals: http://en.wikipedia.org/wiki/Model_theory

 

[D H]

This is incorrect. See http://en.wikipedia.org/wiki/Complete_theory for the relevant notion of completeness and incompleteness.

Show me a proof of the Pythagorean theorem that does not involve multiplication distributing over addition, then.

This is also incorrect. See #18. The informal notion of "power" you have in mind is not the appropriate concept for discussion Godel's theorems.

Any theory based on the reals is, as far as I know, subject to Godel's theorems.

If you want to do physics without talking about measurement you are not really doing physics in my mind.

 

[bcrowell]

Show me a proof of the Pythagorean theorem that does not involve multiplication distributing over addition, then.

Euclid's original proof doesn't refer to numbers or multiplication at all. In any case, it makes a difference whether you're talking about addition and multiplication of natural numbers or of reals; only the latter would be used in a proof of the Pythagorean theorem.

You seem to be asserting that Alfred Tarski's life work is fundamentally flawed. http://en.wikipedia.org/wiki/Tarski's_axioms

Any theory based on the reals is, as far as I know, subject to Godel's theorems.

This is incorrect.

If you want to do physics without talking about measurement you are not really doing physics in my mind.

Nobody said anything about doing physics without measurement.

 

[atyy]

We don't have one yet, so we don't know. But, for example, the standard model has never been stated as an axiomatic system according to the definition used in Godel's theorems. The definition used in Godel's theorems is extremely strict. For example, here is a formal statement of Euclidean geometry: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012 Physicists never want or need to do anything with this level of formality.

-Ben

Yes, but the standard model is not a TOE. My point of view is that there is no TOE. "The physical theory that can be formulated cannot be the final ultimate theory ... The unformulatable ultimate theory does exist and governs the creation of the universe (http://books.google.com/books?id=1f...ook_result&ct=result&resnum=1&ved=0CC4Q6AEwAA)"

But there certainly is a distinguished line of thought that a TOE exists, and hunting it down is the goal of some sub-discipline of physics.

 

[bcrowell]

Yes, but the standard model is not a TOE.

My point was simply that most likely no broad physical theory has ever been formulated as an axiomatic theory in the sense defined in Godel's theorems, and probably none ever will be. (A possible exception is that I did read somewhere that someone had formalized all the propositions in Newton's Principia and worked on checking them with a computerized proof system. Whether this formalization constitutes a physical theory, or just one aspect of it, is a different matter)

 

[friend]

Science seeks an explanation for all things and assumes that all facts are consistent and reasonable. So it is tempting to think that a TOE can be derived from logic alone. As I understand it deductive logic and predicate logic have been proven to be complete. Now if it happens to be that math is introduced as a way to parameterize the spaces used to construct the topologies involved in the unions and intersections of logic, then does the system become incomplete because we coordinatized the spaces involved? Or is it more correct that since the underlying topologies are independent of which coordinates are used, that the math is incidental and should not be used to judge the consistency of the system?

 

[bcrowell]

Here's the book on the formalization of part of the Principia: Jacques Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, https://www.amazon.com/dp/1852334665/?tag=pfamazon01-20 Google books will also let you peek through a keyhole at it.

 

[ordered_chaos]

These proofs of geometry are relative, not absolute proofs. Also, simple systems isolated from interference can indeed be proven consistent and complete. But as soon as things need more robustness, I think that's where Godel's theorem kicks in.

Regardless, physicists seem not give flying fart about it. I do find this strange. They have a lot faith in the language they are using. But I think the closer you get to the truth, the harder it to tinker with less you disturb your experiments.

As physics gets more fundamental, i.e. information theory, I believe these questions surrounding the foundation of math will need to be considered.

 

[bcrowell]

These proofs of geometry are relative, not absolute proofs.

Could you explain what you mean by this?

Also, simple systems isolated from interference can indeed be proven consistent and complete. But as soon as things need more robustness, I think that's where Godel's theorem kicks in.

And this?

 

[yossell]

On the question of whether geometry is subject to Godel's theorems...

I think it's not very hard to effectively embed arithmetic in geometry. Using some fairly simple geometric constructions, you can effectively define + and x geometrically, and then prove incompleteness. I think you need at least two dimensions to do this, but it can be done.

On another issue, I'm just intrigued as to what people think the alternatives to an axiom system is. I don't see axiom systems as abstract constructs of mathematical logic. Euclid had an axiom system for geometry way before the mathematical logicians hit the scene. I just think of axiom systems as an explicit list of the principles that constitute a theory. I would have thought that, without some such list, the theory wouldn't be well defined.

 

[Kevin_Axion]

The thing is that physics in general doesn't require axioms to have a well-defined theory.

 

[yossell]

The thing is that physics in general doesn't require axioms to have a well-defined theory.

Ok - I'm willing to believe it. But I'm interested in how you do get a well defined theory without, at some point, appealing to axioms.

 

[Kevin_Axion]

Calculation and experiment I'm assuming.

 

[yossell]

Calculation and experiment I'm assuming.

I take it we're talking past each other. The methodology of how you discover a correct theory - whether by experiment, calculation, inspired intuition - seems a different question from how that theory is to be expressed and formulated. There's nothing in the nature of an axiom system that prevents it from being arrived at via experiment.

 

[S.Daedalus]

It's quite easy to see that those searching for a theory of everything need not be disturbed by Gödel's incompleteness theorems -- take, for instance, Conway's Game of Life: it's computationally universal, and thus, there are undecidable statements about its evolution; however, it nevertheless has a simple TOE -- its evolution rule.

But really, for anyone curious about the subject, Franzen's book, which I think bcrowell has already mentioned, is the single best resource I ever encountered.

 

[unusualname]

This thread should be in philosophy.

Gödel's theorem is just a result about mathematical logic and meta language, you might as well argue whether physics can refute idealism,

In fact physics theories are even finite in their construction, or can be sufficiently well approximated by a finite algorithm and predict all required physical behaviour, they don't even need the countable infinity of natural numbers.

Physics tries to find a concise instruction set to predict nature, it doesn't say anything about philosophical problems which aren't much more relevant than religion.

 

[Lievo]

Conway's Game of Life: it's computationally universal, and thus, there are undecidable statements about its evolution; however, it nevertheless has a simple TOE -- its evolution rule.

A very concise and clear cut way to show that TOE is possible despite incompleteness.

those searching for a theory of everything need not be disturbed by Gödel's incompleteness theorems

I'm not sure to buy this conclusion however. Incompleteness means that there'll always exist some configuration that Conway Game of Life can reach, altough we can't prove it. Conversely, that seems to mean that someone looking for evidence of a TOE can face data that one cannot prove is allowed by a candidate TOE -even if it's the good one.

 

[Chalnoth]

I'm not sure to buy this conclusion however. Incompleteness means that there'll always exist some configuration that Conway Game of Life can reach, altough we can't prove it. Conversely, that seems to mean that someone looking for evidence of a TOE can face data that one cannot prove is allowed by a candidate TOE -even if it's the good one.

Well, this is just the good ol' problem of inference. In general, we can only prove a theory to be false. There is no observational way to say that a theory is genuinely true.

So while we cannot prove everything in a TOE that falls under Goedel's incompleteness theorem, we can prove some things. And if we prove some things that then turn out to be contrary to observation, the theory is falsified. In the way we typically deal with inference, then, we would progressively gain confidence that the TOE is the correct TOE as repeated attempts to falsify the theory fail, and no alternative TOE that also fits those observations is produced.

Let me state, however, that it may be exceedingly difficult, perhaps even impossible in practice, to falsify a TOE. Our current only existing candidate TOE, string theory, is so far in practice unfalsifiable. I should mention that the mathematical basis of string theory really isn't solid yet. A lot of work has been done, but a lot of work remains.

 

[Lievo]

Well, this is just the good ol' problem of inference. In general, we can only prove a theory to be false. There is no observational way to say that a theory is genuinely true.

Yeah of course, good point.

So while we cannot prove everything in a TOE that falls under Goedel's incompleteness theorem, we can prove some things. And if we prove some things that then turn out to be contrary to observation, the theory is falsified.

However, this is not the solely way we deal with observations. I remember having heard that Mercure orbital motion had only 5% of being as it is according to the theory (I don't remember the details exactly). This was puzzling even though this does not falsify anything. Then one adds that once we take some chaotic feature into account, the probability reach 66%. Far more satisfacting. So I wonder if Godel incompletness allows some generic statistical feature that we can't compute but we would find odd to be apparently unreachable with one candidate TOE. Just a though, I'm not even sure one can axiomatize the question.

Our current only existing candidate TOE, string theory, is so far in practice unfalsifiable. I should mention that the mathematical basis of string theory really isn't solid yet. A lot of work has been done, but a lot of work remains.

Well I ain't no specialist, but seems to me what you really need is data. Go LHC go

 

[bcrowell]

I think it's not very hard to effectively embed arithmetic in geometry. Using some fairly simple geometric constructions, you can effectively define + and x geometrically, and then prove incompleteness. I think you need at least two dimensions to do this, but it can be done.

You've got it partly right and partly wrong. You can certainly define addition and multiplication geometrically, and that's exactly what Euclid did. However, there is no way in first-order logic in Euclidean geometry to distinguish between integers and non-integers. (The restriction to first-order logic is what is meant by "elementary" geometry.) This is why elementary geometry is decidable; it can't encode enough arithmetic to be subject to Godel's theorem. This is what Tarski showed in "A decision method for elementary algebra and geometry." http://en.wikipedia.org/wiki/Tarski The dimensionality of the space is irrelevant.

The thing is that physics in general doesn't require axioms to have a well-defined theory.

I guess it depends on what you mean by axioms. For example, Newton's Principia and Einstein's 1905 paper on SR are both presented in a style that reads like an axiomatization, but they are not formal systems in the sense of Godel's theorem. What one person considers an axiom (e.g., constancy of c), another might label as an experimental fact. I don't think the labeling really has any serious consequences. There are some very important ideas in physics, e.g., the equivalence principle, that nobody has ever succeeded in stating in a mathematically well defined way.

 

[bcrowell]

I remember having heard that Mercure orbital motion had only 5% of being as it is according to the theory (I don't remember the details exactly). This was puzzling even though this does not falsify anything. Then one adds that once we take some chaotic feature into account, the probability reach 66%.

This doesn't sound right to me. General relativity currently passes all solar system tests: http://relativity.livingreviews.org/Articles/lrr-2006-3/

Far more satisfacting. So I wonder if Godel incompletness allows some generic statistical feature that we can't compute but we would find odd to be apparently unreachable with one candidate TOE.

Godel's theorems don't have anything to do with statistics.

 

[yossell]

Hi bcrowell,

I think it's even more complicated than this.

I do agree that the Tarksi's geometry is complete and not subject to Godel's incompleteness theorem. However, in many respects Tarski's geometry is *very* weak. There's no room, in Tarski's formulation, for lines, planes, volumes and hypersurfaces. The variables of his theory range over points, and only points.

This is even more restrictive than insisting the theory of geometry be first order. There are first order theories of geometry where you're allowed lines, planes and volumes - you need extra vocabulary to introduce the relation of one point lying on a surface or a volume, or a line lying within a plane. But you're not necessarily into a second order theory.

Hilbert's axiomatisation of geometry did involve both second order notions, and allowed quantification over lines, planes and surfaces. But you don't have to go that far. Within first order theories that allow you to talk of more complicated objects than mere points, the Godel construction can be done and there is incompleteness again.

Full disclosure: I have some issues with the wikipedia article you linked to. In particular, it makes it look as though Tarski does talk of line segments by introducing the notation xy. Indeed, there's even the confusing claim made that xyCongzw is an equivalence relation. As is said in the discussion, Cong is a four place relation. In the discussion, somebody complains about this, but is (wrongly, in my view) told that one can trivially add new notation `xy' to mean the line segment or the pair of points <xy>. But it's not trivial - from a strictly logical point of view, this is a new theory, which contains a bit of set-theory, and it's not clear whether all the results that Tarksi proved for the original theory - such as its completeness and decidability - will still hold. In particular, if the theory is given too much ability to talk about regions corresponding to certain, first order definable sets of points, arithmetic will be embeddable again, and the incompleteness result goes through.

 

[Lievo]

This doesn't sound right to me. General relativity currently passes all solar system tests:

If I remember well it was about the 3/2 revolution-rotation coupling that was less natural than the 1/1 at first look... I can be wrong.

Godel's theorems don't have anything to do with statistics.

My question is certainly too vague and may completely lack soundness. However if one counts the number of Turing machine that halts as a function of time... this is a statistic and of course this has a lot to do with Godel's theorems.

 

[Chalnoth]

However, this is not the solely way we deal with observations. I remember having heard that Mercure orbital motion had only 5% of being as it is according to the theory (I don't remember the details exactly). This was puzzling even though this does not falsify anything. Then one adds that once we take some chaotic feature into account, the probability reach 66%. Far more satisfacting. So I wonder if Godel incompletness allows some generic statistical feature that we can't compute but we would find odd to be apparently unreachable with one candidate TOE. Just a though, I'm not even sure one can axiomatize the question.

Sorry, I think you'll need to be a lot more specific.

Anyway, in reality, yes, there is always a question of whether or not we have really falsified a theory. Things are rarely cut and dried. The basic reason for this is just that it's actually not feasible to take into account all of known physics when computing the expected result of all but the most trivial of experiments. So instead of taking everything and anything into account, we make approximations. For instance, if we want to know precisely the orbital motion of Mercury as predicted by Newton's laws, in principle we have to take into account not just the Sun and Mercury, but the motion of every massive object in the entire universe. Obviously we're not going to do that, so we make approximations, and attempt to get some sort of estimate of the effect from the stuff we leave out, so that we can gain confidence that our approximations don't impact the final result.

In the end, this sort of fuzziness just comes down to having to be very careful and very thorough when determining whether or not a theory is falsified (in this case, Mercury's orbit does falsify Newtonian gravity quite well, while General Relativity properly predicts its orbit). It doesn't actually impact the Goedel incompleteness theorem stuff at all, because it's just down to experimental rigor and the messiness of reality.

Well I ain't no specialist, but seems to me what you really need is data. Go LHC go

Well, there is that, but unless the properties of our particular observable region are just right, our chances of detecting string theory at the LHC or any feasible collider we have a chance of building in the next few decades is slim to none.

But string theory itself isn't fully worked-out, so if the theorists really dig deeply into the math and flesh it out in full, they may come up with a new way to interpret existing experiment to determine whether or not string theory is accurate, or they may come up with a new but very feasible experiment that we might potentially use to test string theory. In the mean time, lots of work remains to be done just in terms of understanding the theory itself.

 

[bcrowell]

I do agree that the Tarksi's geometry is complete and not subject to Godel's incompleteness theorem. However, in many respects Tarski's geometry is *very* weak. There's no room, in Tarski's formulation, for lines, planes, volumes and hypersurfaces. The variables of his theory range over points, and only points.

This is even more restrictive than insisting the theory of geometry be first order. There are first order theories of geometry where you're allowed lines, planes and volumes - you need extra vocabulary to introduce the relation of one point lying on a surface or a volume, or a line lying within a plane. But you're not necessarily into a second order theory.

Hmm...interesting. Are these first-order theories that include lines as primitive objects decidable, or not?

I think what's becoming more clear to me, both from this post and from the ones about Conway's game of life, is that there's a vast amount of ambiguity in what it would mean to make a formal theory in the Godel sense out of a physical theory.

Here's another example. If you look through Stephani et al., "Exact solutions of Einstein's field equations," you'll find hundreds of examples. Typically they're stated by giving a metric in closed form, and then they can be checked automatically on a computer. There's a heck of a lot of interesting physics in those solutions. I think anyone who knows the physical significance of all of them knows a heck of a lot of GR. In this sense, you could say that GR is decidable. That is, every statement of the form "[foo] is a solution of the Einstein field equations," where [foo] is a metric written in some formal language, can be shown to be true or false by running computer software.

On the other hand, you might want to prove statements about GR such as the Hawking-Penrose singularity theorems. My guess is that any formalism strong enough to prove these is probably also decidable.

 

[S.Daedalus]

Not replying to anybody in particular, but I don't see how one can hope to have a theory of everything be decidable -- after all, there are real world systems, the most obvious ones being ordinary computers, about which there exist undecidable statements. That even occurs in plain old Newtonian gravity: you can build a system equivalent to a universal computer out of finitely many gravitating bodies, and thus, can't predict its evolution in general.

 

[yossell]

Hmm...interesting. Are these first-order theories that include lines as primitive objects decidable, or not?

I believe so. I think one can just take something like the Hilbert formulation, but replace his second-order formulas with first order schema and treat the different variables as different sorts, rather than of being first or second order.

I think what's becoming more clear to me, both from this post and from the ones about Conway's game of life, is that there's a vast amount of ambiguity in what it would mean to make a formal theory in the Godel sense out of a physical theory.

Does this mean that there's a vast amount of ambiguity in the notion of a physical theory itself?

caveat: I may not have understood what you mean by decidable though - the way you use it in the latter part of your post lost me a little.

 

[Lievo]

If I remember well it was about the 3/2 revolution-rotation coupling that was less natural than the 1/1 at first look... I can be wrong.

That is right finally
http://www.nature.com/nature/journal/v429/n6994/full/nature02609.html

 

[Lievo]

Sorry, I think you'll need to be a lot more specific.

Ok let's give it a try. Suppose you have a TOE that perfectly account for any data so far, and your computations predict that either the mater could have dominated (7%) or the antimatter (93%). (your TOE includes some specific prediction which allows you to disambiguate matter from anti-matter).

This would not be strong enough to refute the theory. Still it would be uncomfortable, in the sense that if you can modify your computation to reach 55%, you'll be happy and confident the modification is sound.

It seems that physics includes some computational procedures that are not based on completely firm mathematics (renormalisaton). If I understand the basic reason is we can't compute sums over the infinite when the function does not behave well. This is quite the same as asking the behavior of a Turing machine, isn't it?

 

[Chalnoth]

Ok let's give it a try. Suppose you have a TOE that perfectly account for any data so far, and your computations predict that either the mater could have dominated (7%) or the antimatter (93%). (your TOE includes some specific prediction which allows you to disambiguate matter from anti-matter).

This would not be strong enough to refute the theory. Still it would be uncomfortable, in the sense that if you can modify your computation to reach 55%, you'll be happy and confident the modification is sound.

It seems that physics includes some computational procedures that are not based on completely firm mathematics (renormalisaton). If I understand the basic reason is we can't compute sums over the infinite when the function does not behave well. This is quite the same as asking the behavior of a Turing machine, isn't it?

Sorry, but I just don't get what this has to do with Goedel's incompleteness theorem.

 

[bcrowell]

That is right finally
http://www.nature.com/nature/journal/v429/n6994/full/nature02609.html

But as far as I can tell this has nothing to do with Godel's theorems.

 

[bcrowell]

Not replying to anybody in particular, but I don't see how one can hope to have a theory of everything be decidable -- after all, there are real world systems, the most obvious ones being ordinary computers, about which there exist undecidable statements. That even occurs in plain old Newtonian gravity: you can build a system equivalent to a universal computer out of finitely many gravitating bodies, and thus, can't predict its evolution in general.

I think it may depend on what kinds of theorems you hope to be able to prove about this hypothetical TOE that can hypothetically be made into a formal system. In your example of a computer as a physical system, consider the following two statements that could conceivably be translated into propositions in the hypothetical formal language:

P1. When the computer is put into an initial state S1 at time t=0, its state at time t=1 hour will be S2.

P2. When the computer is put into an initial state S1 at time t=0, it will eventually halt.

P1 is clearly decidable, and it corresponds to a definite prediction that can be tested by experiment.

You might be tempted to say that P2 is undecidable (for generic S1), and therefore the TOE contains undecidable propositions. I think there is both a mathematical objection and a physical objection to this. The mathematical objection is that when Turing proved that the halting problem was undecidable, he did it for Turing machines, but Turing machines have infinite storage space, as well as various other mathematically idealized properties, that may be incompatible with a TOE. The physical problem is that P2 does not correspond to any definite prediction that can be tested by experiment, because no experiment can ever establish that P2 is false.