We frequently see people arguing that Gödel Incompleteness theorems imply that no "Theory of Everything" could be found, because we would expect it to be complete, in the sense that it would adress (at least in principle) any question about the physical world, and consistent.

But the theorems are about mathematical theories and no existing physical theory is completely mathematical. I don't mean that their mathematical formalisms can't be made rigorous, what I mean is that they also have a conceptual background which is not included in the calculational scheme. They are not pure mathematics.

A related question that might shed some light in this issue is: there could be a complete correspondence between physical reality and mathematical objects? Notice, however, that even if this turns out to be true, the question is not settled down, since there is the possibility of our "Axiomatic Theory of Everything" not being expressive enough to formalize the natural numbers and in this case Gödel's hypothesis would not apply.

So, what do you think?

Question

# Do Gödel Incompleteness theorems have any influence on the possibility of constructing / finding a "Theory of Everything" ?

## All Answers (26)

- Nein.
- I recently came across Newton's defense when challenged that his "theory" (perhaps we would now say model) was not causal. Newton replied that he was only trying to describe what nature did, implying that his science of gravity was empirical. I think this evades Godel's incompleteness theorem.

Also perhaps it reveals the desire of a theory for everything as that of a theoretician and maybe something that a science that requires empirical agreement cannot satisfy. - Dear Vilson,

This is an interesting question, though a daunting one. Similar queries, relating to AI (Lucas, Penrose) and to biology, have been articulated in the past, see e.g. Seel and Ladik (1986) “Are we as living beings subjected to Gödel’s incompleteness theorem?”. Unfortunately such inferences mostly led to silence and perhaps even stultification.

The key, as I see it, is to come to grips with, and not to avoid, the problem of self-referentiability and furthermore to understand natural science from this point of view. My view is that this can be carried out properly; the assertion is a bit more serious than it seems here, see also my response to Rajat Pradhan: “What does it mean to say that time can be transformed away in e.g. General Relativity?”.

The crucial idea is not to fall into the Gödelian trap. To exemplify I will return to your question regarding Gödel’s influence on the possibility of having a “Theory of everything”: in string theory the vibrating string is not merely dictating the properties of its host particle – it is being the particle and this requires a “Gödelian” protection which string theory so far does not seem to have. So my answer is clearly YES! - Physics is more than just theory: it essentially involves experiment. Physics without experiment is not physics and string theory is a good example. There is no extant physical theory that doesn't require, at least in its implementation, input of physical constants not derived in that theory. Some physical constants may be derived from others with the help of theory but the essential source of physical constants is experiment. Also some physical theories do obtain physical constant, eg., electromagnetism obtains the speed of light. But even this requires input from the system of units in order to obtain a value. My guess is, correct me if I'm wrong, that string theory doesn't supply the value of any physical constant. So real physics is 2 degrees removed from math, first because physical theory isn't just math and second because real physics requires experiment.
- Dear James,

I think very few disagree with your statement regarding the experimental difficulties to directly verify theoretical descriptions based on string theories. However, my answer referring to the Gödelian conundrum in this connection does not depend on your testimonial at all.

To take another example: Einstein’s gravitational laws of general relativity are experimentally well accounted for – the GPS, the perihelion movement of the planet Mercury etc. Unfortunately there survives unexplained cosmological appearances, e.g. puzzles related to the questions of dark matter, dark energy, the expansion rate of the universe etc. Obviously there is something missing here, which is necessary to identify, e.g. a general (quantum) description of a black hole, which is not presently cured by branes, in order to ”vaccinate” the general theory (gravitation) against Gödel! In this connotation I find Vilson’s question meaningful. - I'm not sure what you're after. Wouldn't you be at least temporarily satisfied by a better theory that still doesn't cover everything? The fact that theories aren't complete in the sense that they don't cover everything they set out to cover seems to me to be an argument that we may never find one that covers everything. The followers of Newton who made scientific capital out of applying Newton's theory (or model) to everything in our solar system were mostly satisfied of its truth for 200 years until Einstein worked out how to make it causal. Some of those scientists knew it wasn't causal but used it anyway. The fact that theoreticians dream of a theory of everything doesn't make it plausible that one exists. It seems to me that the history of physics suggests that one doesn't.
- Dear James,

I agree with what you say, but I find your view a bit pessimistic. Gödel’s theorems in propositional logic can almost trivially be “translated” into mathematical form as a higher order singularity. In other words we can deal with self-references in the more precise language of “mathematics”. There is no physics here so far, however, by a simple relativistic model, using straightforward conjugate variables (operators) it is possible to map gravity to such a higher order singularity (on the second Riemann sheet), which directly leads to Einstein’s laws of relativity, including the Schwarzschild metric etc.

I have explained this in many of my research papers most recently in “Arrows of Time and Fundamental Symmetries in Chemical Physics”, International Journal of Quantum Chemistry 2013, 113, 173–184. I am not allowed to upload it on ResearchGate, but I will be happy to send it to those who want a personal copy.

Best

erkki - Gödelian issues become of importance, not when one is supplying a general framework (which is all that any physical theory does), but when one is attempting to pin down the exact behavior of a specific system: For instance, no universal Turing machine can predict its own behavior to all possible inputs. So no individual can fully comprehend their own mental capabilities--which says nothing about being able to have merely a framework of physical laws.
- Steven,

You are in fact saying that we need to separate biological processes from being describable by physics laws. Can we really do that? Is the emergence of the genetic code such a process? - No, I'm not saying anything about a separation of biological processes from being described by physical laws. I'm saying that describing a biological process--for instance, my brain--requires more than merely laws (physical or biological; it needs data, such as how each neuron connects with other neurons, with what activation potentials, and so on. To describe--predict--how my brain would respond to any possible input, would need complete information on this data, not merely the laws of how things interact.
- Steven,

OK fine. In other words biological processes are commensurate with physics but not enough to define biological processes. Hence one is missing is, in the words of Ernst Mayr, so-called teleonomic processes i.e. those governed by an evolved program. Would the genetic code, how and why, be outside physics? - Um, ain't nothin' outside physics--that's the whole point of physics. Doesn't mean it's possible (humanly) to derive biological processes from just knowing physics, so that's why biologists do things like observe living organisms instead of deriving biological behavior from first principles.

The only way to define any process is to have not only the laws governing its behavior (very few), but also the exact data involved: extraordinarily--overwhelmingly super-humanly--complicated for anything like a living organism. So we can predict behavior only consonant with the level of detail we can nail down, using highly approximate "laws" for that level of detail: "laws" of ecological systems for ecology with data about ecological niches, "laws" about organs for body mechanism with data about how organs are connected, etc.

It's really all just about scale of complexity. Subatomic particles are pretty simple, atoms less so, molecules getting pretty complicated. Stars can be treated as simple objects from some purposes, but their interior organization is important for others; same with galaxies. Scale of complexity for organisms depends on the view being taken: how colonies grow, how organisms feed, how organelles deliver nutrients and wastes, how mitochondria handle energy demands, how RNA synthesizes proteins, and so on. Human ability to collect data is far more limiting than our ability to synthesize laws. - Steven,

Even if I agree a lot with what you say, it sounds a bit pessimistic. The structure of the DNA packing in the cell nucleus as well as the proteins in the cell cycle are governed not only by the data involved but also by the genetic program, usually called teleonomic processes. Would it be unthinkable to also have teleonomy for higher order evolution? After all we have languages, mathematics, etc. to evolve on the social, ecological and perhaps also the cosmical level! - Teleonomy is in the eye of the beholder, no? It means (I take it), "that which looks to be adaptive"--with emphasis on "looks", i.e, this is a subjective description, as it depends on what the observer decides to label as "the goal". Even if the goal is "propagation of descendants", that's a subjective decision to call that a goal; we choose that goal (or some other) because it makes it possible for us to understand the behavior of the system in terms of that goal, but "understand the behavior of the system" is, of course, highly subjective (it's the subject that's doing--or not doing--the understanding).

So, sure, teleon as thou wilt: If you can spot what appears to be a developmental agenda for, say, human societies, on the order of centuries or millennia, then that might well be a useful organizing principle that furthers the understanding of historic development and be speculatively predictive for the future.

Nothing pessimistic in my view of things: We think further than those who shoulders we stand on, observe more, and come to finer conclusions. Lookin' good, I'd say. - In my book (2005) I wrote, dear Vilson, that the dilemma of Cantor can only be resolved via Gödel incompleteness: i.e. by crossing the axioms' borders (limits) of set theory and by bringing physics (nature's observation) into the game. The solution of the dilemma of Cantor was crucial for us to formulate the TOE on the basis of Cantor's Continuum (as descriptive of the vacuum). So, the answer to your question is imho: yes.
- What is the dilemma of Cantor? There's a novel by the name of "Cantor's Dilemma," but that's about scientific ethics, not metaphysics.

The continuum isn't due to Cantor, it's Dedekind who defined it. And all it is, is a mathematically handy model for the universe. As far as scientific experiment goes, the universe might just as well be modeled by the rationals, not the reals--it's just a lot handier to have the continuum available for such things as trig functions, square roots, etc. But there's nothing truly physical about the continuum. No reason you couldn't base physics on the rationals. - Dear Steven, thank you for your question. Cantor's dilemma stands for his inability to set the first two infinite cardinal numbers (of infinite sets of natural and reel numbers) into numerical relation. It was one of the century problems formulated in 1900.
- Cantor may have been disturbed by the relation between the cardinality of N (the natural numbers) and that of R (the reals), but the "dilemma" has been "resolved" for half a century now: As shown by Cohen, the usual axioms of set theory allow one either to assume that there are no cardinals between those two, or that there are cardinals between them. It's not related to Gödel's Incompleteness Theorem, except in that it shows another sort of "incompleteness"; but it's a very different (and, I think, far less troubling) sort of incompleteness, in that it applies solely to Zermelo-Frankel Set Theory and the question of existence of cardinals, not (as with Gödel) any axiom system at all that allows for the set N and the question of the existence of a proof for any true statement.

It's typical for mathematicians to assume that there is no cardinal between the size of N and the size of R; but I think that's merely because we don't seem to have any "practical" need for intermediate cardinals, so things are simpler if we just assume them away. I can't imagine how either physical observations or physical theories might influence this choice of axiomatic assumption, which seems to have only mathematical implications (and those of a highly abstruse nature, even by the standards of mathematicians). - Sure, dear Steven, Cantor's dilemma and Gödel's incompleteness are causally/historically not related. Yet, imho the incompleteness of Gödel can be taken as argument (as I humbly have done) to cross the limits set by mathematical (Zermelo-Fraenkel) axioms and to look at nature (observation of nature = physics) in order to resolve (not "resolve" as you say has been done by Cohen) Cantor's Dilemma. But you are the expert and I am happy to listen to your arguments.

As to the necessity of knowledge about the cardinal numbers I may mention my humble conviction that the natural constant of the speed of light (which is the only "big" natural constant) is related to the first cardinal number of the continuum. For this reason, I carry the opinion that the knowledge of/about cardinal numbers is crucial to understanding nature and vice versa. - Hmmm, I can't see any sense--except the purely human--in which the speed of light is big. It all depends on the units one chooses; for instance, in the units--the so-called geometric units--favored by theoretical physicists, it's precisely 1 (i.e., light travels one lightyear per year, or one light-second per second). It's only human-scaled units (like seconds and meters) that yield large numbers.

I don't see a connection between hypothetical cardinals intermediate between the size of the natural numbers and the size of the continuum, and anything we can observe in nature. Experiment, by its very nature, is totally insensitive to anything about the infinite, as our observations are inherently approximate. (Even the "infinite universe" models constructed in relativity theory and used as test-beds for our understanding of what observations tell us about the universe, are nothing more than highly simplified models chosen for their mathematical ease of use, not because observation could ever conceivably show us that the universe is infinite.)

Which isn't to say that it's impossible to construct models for the universe that make use of intermediate cardinals--though I've no idea what such a model would look like. - I understand your reservations, dear Steven. The point seems to be to find a natural but abstract framework in nature that can numerically be described. It is my humble opinion that it does exist for reasons of stability of nature. Only sound logical descriptions of natural facts and events could imho ensure steady rigidity. What is more logical than mathematics?
- Dear Steven.

I like your way of seeing the world!

I will detail some understandings that resulted from my work.

I believe this is the final theory that Einstein dreamed!

First I would like to note a few points.

What is a meter?

What is a second?

What is a pounds?

The answer to these questions is very simple, arbitrary units are chosen for humanity to relate to its surroundings.

The second aspect which is the speed, if not an arbitrary number based on these measures, which expresses a relation but this one working with a value!

Here we can observe that the invariance of the speed of light and truth to the invariance of space-time relationship in an inertial reference and nothing more than that!

And the Lorentz transform, demonstrates how these relate inertial frames between them.

Starting from the idea that the speed of light and fixing born the theory of relativity.

That is correct for a vision of the inertial frame.

However if we look at the universe from outside the speed of light and variable.

And own Lorentz transform allows you to see the variation of this speed.

If we take a privileged referecial static to measure all points within the universe the speed will always be different in all of them.

That is the speed of light and variable continues, and at the same time.

Only depending on the vantage point that is taken as a reference!

Following this line we have the quantum mechanics where time does not exist and expaso. But it is within the same universe as the theory of relativity so it has a relation with the same!

The quantum mechanics has two possible representations.

One comes from the Schrodinger wave equation and one in Hilbert space.

In the case of the wave equation she works with complex numbers and many solutions become very complex if not unattainable to be calculated!

Since the Hilbert space where most observe the quantum mechanical calculations.

At this point and interesting to observe that the Hilbert space is infinite, complex matrix and vector extruded.

Ie it works as a support for the particles enter and analyze their relations without time or space since the matrix all the points are interrelated!

Ie quantum mechanics and the theory of finite elements of another theory!

Where details what is not and the element but only the relationship between the elements they teem!

And here comes my job demonstrating that everything originates from a very small and dense string.

This string when a lively movement that is the essence of all energies, creates the universe!

In this definition the universe and closed!

E is defined as a massively parallel quantum computer.

Where the string and the element of unitary quantum processing.

Creating interference figures is in fact a holographic film.

We call the Universe!

Finally I managed to set everything in physics with only 4 numeric values that emerge from this string and your move!

And I posted a job on how to derive the theory of relativity... [more] - Dear Jorge,

I have very little I can contribute to the notions of combining quantum ideas with relativity. I just note that the original quantum ideas were based in an essentially Newtonian background, totally ignoring anything about relativity; and that the only really successful mingling of that original quantum conception with relativity is in Minkowski space, i.e., totally flat: no gravity, no matter fields, no curvature. There's been an enormous amount of "quantum gravity" propounded, but absolutely none of it is well-founded; in essence, nobody knows how we can combine quantum ideas with gravity--even though a great deal of effort has gone into trying to say something about quantum processes in gravity or curved spacetime.

Hilbert space of annihilators and creators works really well for empty, flat spacetime--but otherwise? No one knows how to do it. Same with Schrödinger equation.

And I can't even say what the difficulties are; that's not within my strengths. - Dear Steven

The problem is even deeper than the Newtonian vision .

But the problem and the source of modern physics.

All theories of physics teem based on the concept of field .

But no one can tell what a field!

And it was just that I worked on and found a model that allows to describe the construction of the space , which is the basis for the propagation of waves.

Ie the space is the field in which Maxwell's equations interact .

What I would like your opinion and expert in mathematics on the fact that this model allows algebraic equations to have all the physical constants and allow to manipulate these equations within the normal equations of physics getting simpler algebraic equations in the domain and are based on 4 values and yet be absolutely correct when compared with standard models .

The question is an algebraic equation to be simplified with another equation and the result be correct in my understanding is correct because it is the mathematical point of view.

This correct my vision ?

Please review the work that will appends the mathematical point of view like you ve ?

Thank you. - I'm afraid I can't offer any feedback on calculation of constants. In the first place, I've no idea where values of physical constants come from--indeed, I'm not convinced that question even makes sense. In the second place, I don't see theory lying behind your calculations, so there's no way I can express an opinion on the theory.
- Dear Steven.

The model is part of a string, with a diameter and length are fixed.

This string assume different geometric shapes, as the energy it receives.

Having 5 geometric shape possible.

These 5 ways are divided into 2 forms that construct the space and time.

One form is the basis to build the cluster and sub-atomic particles.

And the remaining two forms are one occurs in situations of uniqueness and the other is a transient form that transmits waves in the fabric of space created.

In this model we have 5 dimensional.

Three spatial and two temporal.

The time are the result of the transfer of information between two points of the space created.

These waves and one transverse and understand how the speed of light.

And the other, and a longitudinal wave and understand how the speed of quantum entanglement, which is trillions of times faster, than that of light.

The overall understanding of this model and a massively parallel quantum computer that mounts alone shall.

Resulting in a fluid filled structures with uniformly distributed fluid that moves the blades as membranes spherical structure.

Where one side of the "brane" to this matter and the other antimatter.

And the movement creates interference figures that generate the apparent reality in which we live.

A holographic universe in memory of a large computer.

The string is projected this universe we live in as the magnetic permeability and electrical permittivity of space.

What are the two numbers that are the basis for all algebraic equations that result from the operation of the machine.

The other two values that are used in the equations of relativistic corrections velocity and local density.

In this model the time and only delay the transfer of information between two points in space.

And this model for all the constants of physics has an algebraic equation that uses the projection of the string to generate the values that physical measures today.

As the mass of an electron, the electric charge of the electron, the Planck constant-the constant newton and all others.

Over can be used in the normal equations of physics and subsequently manipulated algebraically and turns any physics equation and equation based only on the string and much simpler normally apply the equations that we have nowadays.

If you look at the works you will see that for example the equation of relativity Einstein although simple is further simplified with reference to this string.

I hope to give a little information about the work I did, and well he has four years of extensive equations and hundreds of pages of calculations.