Impact of Gödel's incompleteness theorems on a TOE

PhysDrew

One more thing I would like to propse which was brought up earlier, is what the everything in the TOE actually means? I've heard it discussed that a TOE may have predictive power, but however I have also heard by other physicists that the TOE will be nothing more than a joining of mathematical realtionships to give one unique 'formula' with little to no predictive power, rendering it to low importance in the grand scheme of things. I think Godels theorem/s may or may not apply depending on which scheme it is applied to?
By the way has anyone noticed the self referential hint I put into the title?!

Fra

I wasn't sure wether to bother entering the discussion, because maybe it's not possible to be brief so maybe I should keep quiet, but I'll just throw in my wet socks and be done with it.

If you believe in that information is bounded and that any observer process and hold only finite info, there is a limit of the amount of confidence any information processing agent/observer can accumulate. At some point there is a saturation and you can't inflate the theory by adding the gödel scentences as another axiom, where the progression ends, and all the TOE means is an effective basis for further actions. Also there is not guarantee that there is any convergence to a unique TOE, it may be that it just keeps evolving by destruction of axioms and generation of new ones.

I'll just make the story short and att my OPINION that I think an analysis of all this, decidability, inference, incompletness theorems support by plausability (but not imply in any logical sense) the idea of evolving law and that it's impossible to distinguish cleanly between law and state, simply because they are both merely results of inference, and the limit of infinite confidence in a unique limit seems seems highly unphysical.

If you think about the limits of a TOE, it seems to be at least that is subject to constraints similar to the information state. The TOE represents the observers "state of information" of the expected laws of evolution of the information state. So it's another layer of information.

The more common idea that laws of physics are eternal, correspond as I see it to the "infinite confidence" limits in the progression mentioned. But this limit may not exists, for two reasons (unknow non-uniqueness/convergence) and the lack of information capacity to encode this limit/confidence (if at hand).

This can be expanded alot, but I agree it does enter philosophy, and it's also not very conclusive therefor I think it's value is mainly suggestive and inspirational. Therefor there is a limit to how much it makes sense to elaborate. This is already a long thread but i've added my short input.

/Fredrik

Hurkyl

I'm stating that Tarski's system is not as limited as you make it out to be. There are clear limitations -- e.g. that the theory of integer arithmetic cannot be expressed in it -- but it is not as limited as you claim -- e.g. it is able to discuss lines.

The objection you had against lines was that you were rejecting the syntactic tools of first-order logic. Constructing the type of Euclidean Lines from the type of Euclidean Points, for example, is pure first-order logic -- you are not creating a new theory by doing so.

Chalnoth

Yes, this is pretty obvious.

Basically our only hope of finding the TOE is if it turns out that mathematical consistency severely limits the possible TOE's so that the correct one can be experimentally distinguished from the others. There is obviously no guarantee that this is the case. And there is certainly no guarantee that we will be able to genuinely demonstrate that there aren't any TOE's that are also consistent but experimentally indistinguishable.

Well, you can make any laws of physics eternal simply by having the laws of physics describe any changes that occur. But in any event, obviously infinite confidence isn't possible, and I wasn't attempting to imply it was.

bcrowell

Like Hurkyl, I'm not convinced by your assertion, and I haven't seen you offer any evidence for it. My #38 does offer evidence for my assertion, in the form of a reference to Tarski's paper "A decision method for elementary algebra and geometry." Anyway, I don't know if any of this even matters for the purposes of this thread. It seems that we all agree that there are weaker and stronger formulations of geometry, that the weaker ones don't give enough arithmetic for Godel to apply, and that the stronger ones do.

bcrowell

I agree disagree only because IMO this is putting it too mildly. I don't think there is even any clearly defined notion of what it would mean for a physical theory to be decidable.

And again I disagree only because IMO this is putting it too mildly. I think it's even less than a quibble over what constitutes a TOE. If we get a TOE, we'll know it's a TOE because it will unite the four forces, reconcile GR with quantum mechanics, and make testable predictions that are verified by experiment. That's how we'd know what constituted a TOE. We can't use decidability to define what a TOE would be, because the notion of decidability is fundamentally inappropriate for talking about physical theories.

Lievo

Which states that the tape is unbounded, and that's not the same, technically speaking, as infinite.

This is just making no sense! Ok two ways to see it:
-the tape of any TM that halts is bound by the busy beaver function. It's large, it's unbounded, and it's finite. If the TM doesn't halt, you don't need any tape at all, because you'll never get the result of the TM -which is define as what remains when the TM stops. Of course you can't, in general, know if your TM belong to one or the other class. Doesn't change the tape you need is finite.
-if you think it's infinite, then precise what infinite you're talking about. Is it aleph 0? No, or you'd enter hypercomputation. Is it more? No, that's worse! Is it less than aleph 0? Well, if you believe there is such a thing as an infinite less than aleph 0, go and publish!

Sorry, I'm begining to be fed up of stating the obvious. Last time I comment this.

D H

Technically speaking, saying that the length is unbounded exactly the same as saying it is infinite.

Chalnoth

If you don't know whether or not the program halts, then you don't know beforehand how much tape is required, which means you need infinite-length tape or risk your computer crashing.

Lievo

Chalnoth, as you previously teach me some stuff I should not refuse to answer you.

What you need, mathematically speaking, is at most the busy beaver corresponding to the number of non blank in the initial state. Nothing more, and that's finite. Pratically speaking, be sure your computer will crash before reaching it.

D H

Baloney. Busy beavers halt. Halting is not a requirement of a Turing machine program. An algorithm to compute the decimal representation of an irrational computable numbers such as the square root of 2 will not stop. The decimal representation of course requires an infinitely-long tape.

Coin

We are getting a little off topic. But.

I think some of this disagreement might be avoidable if we speak precisely.

1. A turing machine must have access to an unlimited amount of tape in order to capture the full power of the turing machine formalism.

2. Any given turing machine, when run on an input for which it terminates, will only use a finite amount of that tape.

The lack of a limitation on space can't be waved away by saying "but that's only for terminating programs", because a significant portion of the power of turing machines in the first place comes from the fact that turing machines have the option of never terminating. The decidable languages are a smaller set than the recognizable languages. If there is a known bound on either runtime or space used, then all programs are "decidable" and you have something weaker than a turing machine.

You are correct that the busy beaver function does define an upper bound on space usage for halting programs. However this statement is not useful to know, because BB(n) is noncomputable. (In other words, BB(n) is a bound, but it is an unknown and unknowable bound). In fact your statement here is basically tautological, because BB(n) is defined as the maximum number of tape squares used up by an eventually-terminating program of a given complexity. So what you are saying is that the number of squares used by a terminating program will be equal to or less than the number of squares it uses.

Do you disagree?

friend

A Theory of everything would at least explain why there are particles, fields, and even spacetime to begin with. Are we going to be fully satisfied if we predict and/or discover smaller, higher energy particles? No, we'll wonder where those came from, and so on, etc, etc. I think we will not be satisfied until we have explained everything in terms of the principles of reason. Once you derive physics from logic alone, then what is there left to question? There would be nothing left except to maybe question your sanity. But to predict the properties of things does not explain where they came from.

I think a theory of quantum gravity would probably be a TOE, because it would unit QM with GR, particles and fields with spacetime. And to unite spacetime and particles fields would probably require a theory from reason alone. For physically, there is nothing more fundamental than spacetime and particles/fields. And to explain something more fundamental than what is physical would have to rely on complete, abstract, generality about anything true; it would have to rely on principle alone.

So here we are discussing whether physics can be reduced to a complete axiomatic system. The question is not even relevant unless we can derive physics from a system of reasoning. I don't think it will reduce to the axioms of geometry because we would still have to explain why there is geometry, or spacetime, to begin with. It would have to reduce to reason, or we would still have questions and it would not be complete. (am I using "complete" equivocally here?)

Chalnoth

The difficulty is that there is no guarantee that there exists only one unique TOE. There may be a great many potential ones. Even after discovering a potential TOE, we would still need to determine whether or not it applies to our reality. Pure rational deduction can never ever tell us whether or not a TOE applies to our reality.

D H

You are using complete incorrectly here. This is a thread on Godel's incompleteness theorems, and whether they have any relevance to a TOE.

Lievo

I should not refuse to answer anyone. My apologies for this misbehavior.

In case you care, please notice there is no such thing. Precisely because you can't compute something that need an infinitely-long tape. You may wish to read wikipedia on this.

...I hope this is not quote mining too much.

Well said (...) No, except for some details: decidable and recognizable that's the same, I think it's usefull to know that BB(n) is the bound because basically, this is what separate computing from hypercomputing, and finally the fact that the bound is uncomputable doesn't make it an infinite. But I begin to desesperate here.

yossell

Don't know what you've got in mind when you talk about the `construction of types'. The existence of lines is not a first order consequence of the existence of points. The existence of first order definable regions is not a consequence of a theory that only talks about points. I Can see what you might have in mind if you've got first order set-theory in the background. But that is to add something to Tarksi's theory.