# Is there something similar to Gödel's incompleteness theorems in physics?

Gödel's incompleteness theorems basically sets the fact that there are limitations to certain areas of mathematics on how complete they can be.

Are there similar theorems in physics that draw the line as to how far one can get in physics as far as completeness?

• Sorry for the lack of good tags, didn't know what I should put. – Simply Beautiful Art Jan 15 '17 at 23:19
• Related/Possible duplicates: physics.stackexchange.com/q/14939/50583, physics.stackexchange.com/q/156909/50583 and their linked questions. – ACuriousMind Jan 15 '17 at 23:43
• The Heisenberg Uncertainty Principle draws the line as to how far one can get in physics as far as completeness of information about a system. – Michael Jan 16 '17 at 5:27
• If you have not yet you should read this paper by Stephen Hawking hawking.org.uk/godel-and-the-end-of-physics.html – Not_Here Jan 16 '17 at 6:31
• How similar do you want it to be? If e.g. parallel universes exists, we likely can not interact with them. But since physics is about the things we can interact with, and can prove, it could be argued that anything outside that is not physics anymore. – PlasmaHH Jan 16 '17 at 10:09

No, there is not nor can there be a similar statement in physics. That is because we can know all there is to know about the mathematical systems we construct; after all, we have set them up ourselves (but then Gödel's incompleteness theorem tells us that there can be features of certain systems that remain unknowable; sorry for butchering what the theorem really says, btw).

Physics, on the other hand, ultimately attempts to model reality. The problem there is that we have fundamentally no way of knowing reality in itself; we cannot even be sure there is such a thing as reality although we take that as a fundamental axiom of physics. Thus, all we can do is propose models for the reality we experience. We cannot know what the relationship of such models to reality as it is might be.

• I think the last paragraph may actually be an answer to the OP's question. I find that the idea that science cannot help us know anything about reality is on part with the level of limitation Godel's theorems put in place. I also find it to be similarly revolutionary -- it's amazing how many people are really uncomfortable with it! – Cort Ammon Jan 16 '17 at 1:58
• I feel like it's also key to point out that Godel's incompleteness theorem only says that for a particular set of consistent axioms that are sufficiently powerful, there are necessarily things in that system that cannot be proven. But this doesn't mean that there isn't some more powerful set of axioms where some of those things CAN be proven. – Shufflepants Jan 16 '17 at 15:10
• What you write is in line with some ideas Bohr outlined about QM and is also within Kant's idea of the noumena and phenomena setting. However, for classical computing systems Turing's results give limits on algorithmic computation. In some sense we might say Godel's theorem applies in a basic classical sense as a result. – Lawrence B. Crowell Jan 16 '17 at 16:37
• @Shufflepants: I'm pretty sure that it's turtles all the way up, and that even if you find some 'more powerful' sets of axioms, there will still be true statements within that system that cannot be proved by those axioms. – whatsisname Jan 17 '17 at 0:37
• @Cort Ammon: It's also why I think that dogmatic materialism is just as wrong as dogmatic religion, as well. Declaring that everything real both known and unknown ultimately must reduce to matter is a sweeping, sweeping statement. It also makes me think of some of the advantages of alternative knowledge theories like "Feminist Situated Knowledges" (Haraway, etc.) where we have no knowledge of any "objective" reality but only knowledges localized to a particular situation or context. But this is all philosophy now, not physics so we're really kind of going off topic here! – The_Sympathizer Jan 17 '17 at 3:59

I am going to first change this question a bit. Consider quantum states as quantum information or quantum bits processed by Turing machines (TM) governed by Hamiltonians. We may think of the erasure process of removing symbols in a Turing machine move as recovered with auxiliary registers, so irreversible processes are a problem that can be avoided. I would say it is better to ponder whether exhibit the algorithmic incompleteness proven by Alan Turing, that there does not exist a universal Turing machine (UTM) that can determine the halting status, or as it turns out a number of other features, of all possible Turing machines, applies to quantum mechanical evolution. So we have some Hamiltonian, if the system is Lie algebraic this is given by the product of roots that act as raising and lowering operators, and the system evolves in a textbook fashion. Can there be some sort of algorithmic incompleteness with this?

I am going to first off say that most physicists either shrug their shoulders or they might actually get rather “testy” over the proposition. Most physicists think rather not. Of course if the majority of physicist think this and you have nothing else to do, then by all means at least ponder the possibility! I will also say as a second opinion in this paragraph that frankly I have no idea for certain about this, but why not at least think about the possibility? The worst that can happen is that I am wrong.

Where might this incompleteness occur in physics? I would say one possibility is with quantum measurement. Quantum mechanics is perfectly deterministic and it computes the evolution of amplitudes who's modulus square give probabilities of outcomes in a measurement. However, we have no theory for how an outcome actually obtains. There is no dynamics for this, and attempts at this run afoul of Bell's theorem and other limitations of quantum mechanics. Yet nature produces an outcome! Quantum interpretations have holes in them and effectively reduced quantum mechanics to metaphysical categories that fall short. We may think of the measurement process as a set of quantum states being measured by an other set of quantum states, usually far more quantum states, and in the end this is a sort of self-referential loop. This is similar to a UTM that emulates other TMs, or a predicate that acts on Godel numbers for predicates in Godel's first theorem.

To give a possible physical case of a system let us consider the Reissnor-Nordstrom metric of black holes (BH). The Penrose diagram below illustrates null geodesics entering the BH, which pile up near $r_+$. Suppose that in the exterior region there is a Turning machine that computes a non-halting problem. The infalling observer in an eternal black hole in principle detects the computation in a finite time period. The infalling observer or computer may be a universal Turing machine that determines the halting status of any possible Turing machine. This is a Malament-Hogarth (MH) spacetime that is as a hyperTuring machine able to solve uncomputable problems. A BH in principle absorbs qubits and permit interior observers to deduce whether any problem will halt or not. This argument holds for an eternal BH, while in reality BHs emit Hawking radiation and are not eternal. Also such BHs have complex quantum hair. The diagram has been adulterated to illustrate this where the BH has a finite duration. Consequently a BH with hair will not be able to determine if all possible Turing machines halt, but it will be able to determine if a significant number will. This will adjust the Chaitin halting probability related to the Chaitin constant. Whether a Turing machine can halt or not is given a probability not universally computable. Consequently the dice have been favorably loaded in some unknowable way to favorable to decide halting status. A physical hyper-Turing machine is a truncated version of the ideal.

Potentially Godel's theorem has some relationship with consciousness. Douglas Hofstadter wrote an entertaining book $\it Godel~Escher~Bach$ that explored the idea of consciousness as self-reference. Goedel's theorem and Loeb's theorem permits unprovability to be cast in modal logic, see Boolos Burgess and Jefferies “Computability and Logic.” For $\square$ meaning necessarily and a proposition $p$ then $\square p~\rightarrow~p$ is true, but Godel's theorem indicates $\exists p:p~\rightarrow~\neg\square p$. This is a counter example to the argument Anslem gave for the existence of God. This means that a proposition that is a fixed point of some predicate built from provable and true functions is equivalent to a functional combination of false statements. This means that in a modal sense that $\neg\square\neg~=~\Diamond$, which means possibly, indicates a sort of “freedom” that exists in mathematics. In the sense of computation a system, such as a truncated hyper-Turing machine, may estimate the truth value of propositions according the Chaitin's number $\Omega$.

It might be that consciousness is also a truncated hyper-Turing machine that approximates the ideal of a completely self-referential system that can “jump out of an algorithm,” or make a leap of imagination. A truncated system may be able to perform these actions, but not in a complete “God-like” form. An ideal hyper-Turing machine is able to perform “trans-provable” operations, which can include choosing between unprovable “axioms” in order to construct a model necessary for the function of that system. For a physical system the system is not perfect, and at best this can operate under the bounds of unprovable Chaitin probabilities. There is then a relation $\Diamond~\leftrightarrow~\Omega$ which operates within these bounds. The fact that this involves $\Diamond$ or possibility means that from a physical perspective there is a relative entropy of states associated with this uncertainty.

This touches on the physics of quantum gravity, and in many ways I have thought the questions concerning wave function decoherence in Hawking radiation had connections to the measurement problem. We might then ponder where this comes in with math. The quite possibly the Freudenthal system of triples of $E_8$ or ${\cal O}^3$ might be a structure underlying string theory. This contains the $26$ dimensional bosonic string, and also contains the Leech lattice. The Leech lattice or the sporadic Mathieu group ${\cal M}_{24}$ is the automorphism of the Fischer-Greiss “Monster” group. This in turn has been found to have implications with number theory, called moonshine or umbral moonshine. My black dog I named umbral. Now we can then see how in some subtle way in the mathematics that Godel's theorem might rear it head.

So this is rather speculative, and I know there will be those not happy with this. However, people who follow the rules and always do as told rarely turn up in history.

• Do you have a reference for your claim that the RN black hole spacetime works as a "hyperTuring machine"? – Danu Jan 16 '17 at 11:21
• Try this one arxiv.org/pdf/gr-qc/0104023v2.pdf , which works with the whole Kerr metric. The Kerr metric is difficult under any circumstances. – Lawrence B. Crowell Jan 16 '17 at 12:58

For a long time, I've been fascinated with a set of theorems that seem to define the boundary of the knowable universe. And by that, I don't mean just the limit of what we know today, to be extended as we learn more tomorrow. I mean the absolute limits of science and reason, beyond which we can never venture, no matter how clever we are. These boundary meta-theorems include the following:

• Heisenberg's Uncertainty (Physics) — There is a limit to how accurately we can measure the properties of physical objects.
• Bell's Inequality (Physics) — That limit applies not just to our ability to measure things accurately, but to our fundamental ability to know things about physical objects.
• Gödel's Incompleteness (Mathematics) — Any attempt to explain everything using a small(er) set of axioms is doomed to be either unfinished or wrong.
• Turing's Undecidability (Computing) — There are infinitely many problems that cannot be solved by any digital computer.
• Chaitin's Irreducibility (Computing & Mathematics) — Almost every number (probability = 1) is "random" in the sense that it cannot be computed by an algorithm that is much shorter than the digits of the number. That is, the shortest name for the number is the number itself. Randomness exists in mathematics as well as physics!

Notice the last word in each of the theorem names. Each of them expresses a negative. Each of them tells us something about what we cannot do, where we cannot go, what we cannot, under any circumstances, know. Intuition suggests these five principles, despite their different fields of application, are somehow related. In fact, they seem to be exactly the same, or at least stem from the same underlying phenomenon. Namely this: Fundamental randomness exists. It isn't just a small wart on our logical world, but rather an unfathomable ocean surrounding it. We cannot ever know what goes on in this ocean of randomness. We can only glimpse the shape of the coastline of our little island of reason, as the theorems and principles listed above begin to illuminate.

Specifically for physics, Heisenberg's Uncertainty Principle basically says, certain pairs of properties of physical objects — simple things like where it is and how fast it's going — cannot be simultaneously measured with perfect precision. The more carefully you measure the position of, say, an electron, the less certain you can be about its velocity at that same moment. If you are very, very, very careful measuring the position, then whatever number you observe for the velocity is essentially meaningless; it is random beyond a certain number of decimal places. Now this limit on how accurate one can be with these combined measurements is quite negligible for larger objects like bowling balls or BBs, but for small things like electrons and photons it makes a difference. The combined limit on our accuracy of measurement is determined by the reduced Plank constant which is about 35 decimal places of accuracy. Beyond that, physical properties are universally un-measurable.

HUP can be understood by thinking about how one's measurements affect the object being measured. Measuring the position of an electron involves shining a light on it, and a more accurate measurement requires shorter bandwidth, higher energy photons. When the electron is impacted by the high-energy photon, its velocity is affected, thus introducing randomness.

And that is the way it was presented and talked about at first, as a limit on experimental accuracy. The Quantum Physics textbook I used in college, the 1974 edition of Quantum Physics, by Eisberg and Resnick, explained the Uncertainty Principle by saying, "our precision of measurement is inherently limited by the measurement process itself [...]." Albert Einstein, and many other prominent contemporaries of Heisenberg, believed there must still be an underlying set of "hidden variables" that control the universe and provide precise, deterministic answers to any question, even if we were forever limited in our ability to experimentally verify those answers due to the Uncertainty Principle.

Einstein, together with his colleagues Boris Podolsky and Nathan Rosen, even wrote a famous paper in which they, almost mockingly, proved that Quantum Mechanics must be wrong, or else the world as we know it would be a truly strange place. To do this, they assumed only two seemingly obvious things about the world. First, that objects have intrinsic properties like position and velocity, even when no one is measuring them. This they called "reality." And second, that measurements of reality in one place and time cannot instantaneously affect other, far away realities, a property they called "locality." Einstein, Podolsky and Rosen basically said, who would want to live in a world where reality and locality did not hold. In other words, they believed our friendly, orderly universe could not possibly be intrinsically random.

But they were wrong.

In 1964, Professor John Stewart Bell proved a result that some have called, "the most profound discovery of science." The unassuming title of his brilliant paper, On the Einstein Podolsky Rosen Paradox, referred back to the "paradox" outlined by Einstein and his pals. Bell proved that the universe is in fact fundamentally, inherently, inescapably random. More precisely, he showed that no deterministic theory based on hidden variables could possibly explain all the observed results of Quantum Mechanics. And if that means there is no such thing as reality or locality, then so be it. Either the principle of reality or the principle of locality (or both) does not apply in our universe! A strange place indeed.

Heisenberg's Uncertainty Principle is not just a limit on how accurately we can measure things. It's a limit on what we are allowed to know about the universe in which we live. There are physical quantities that are universally unpredictable. At the very foundation of our familiar physical world, lies the Divine Random.

• I would like to point out that the first two topics are clearly related to "the universe happens to like the group of scaled rotations", the second two are different forms of the liar paradox, and the last says that $2^N \ne N$ even when $N$ is an infinite set, which is a "diagonal argument" and hence perhaps a generalization of the liar paradox. – CR Drost Jan 16 '17 at 17:47
• Whether objective randomness (or probability at all) exists is an open question and has been receiving a great deal of dispute. What the statements you enumerate here have in common is not randomness, rather it is self-reference. After all, there is nothing random in Gödel's incompleteness theorems. – Zetaman Jan 23 '17 at 22:15

Mathematics is language pushed to the limits of objectivity and precision, where all concepts are clearly defined and interrelated, and their articulations are governed by explicit rules. In that context Gödel's theorem states a fundamental incompleteness of language itself.

As long as we define physics by what we know about the world, it is submitted to the same limitation: language will never be complete enough to describe the world, because language in its very essence is intrinsically limited.

Now physics is more than the elaboration of a description of reality: it is an actual interaction with reality, at more and more subtle and deep levels. There is no indication of a hard limit to that endeavour.

Yes, certain "incompleteness theorems" do exist in physics. The one I know about is the "undecidability of the spectral gap" in quantum many-body physics. A recent preprint as well as a refereed paper in Nature describe the situation more completely. Reproduced below is the abstract to the paper in Nature, which gives a better summary of the proof than I could presume to do.

The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum many-body physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum many-body system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a two-dimensional lattice with translationally invariant, nearest-neighbour interactions, for which the spectral gap problem is undecidable. This result extends to undecidability of other low-energy properties, such as the existence of algebraically decaying ground-state correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

There is a question of 'how much we can know in physics' which is currently bothering the folks who study quantum gravity. I heard about this from a lecture by Nima Arkani-Hamed (which you can find here), and the idea is roughly like this:

In relativistic quantum mechanics, probing phenomena with a high characteristic energy scale is equivalent to probing short length scales. Thus, experiments like the LHC attempt to pack as much energy as possible into as small a region of space as possible, so as to find ever more massive (i.e. energetic) particles.

The energy scales associated with gravity are so high that nobody ever hopes to explore them in a particle accelerator. However, there is an even more fundamental conceptual problem with probing quantum gravity, which suggests it may not be possible to explore certain quantum gravity effects even in principle: In order to reach the necessary energy density for seeing quantum gravitational effects, you would have to put so much energy into such a small region of space that it would collapse into a black hole, and thus you ostensibly wouldn't be able to get any information out of the process anyhow.

This is an active matter of research, and not at all well-understood, I would say. However, it is an interesting parallel to Gödel's theorems, but with a more physical twist: The argument does not say anything about how much we can learn about physics from analyzing its mathematical structure, but rather how much we can learn from that perhaps even more fundamental tenet of physics: Doing experiments!

All physics does it model observations in terms of mathematics. So of course, it could turn out that some theorem in say realativity is not provable because of Godel incompleteness. And that could have some reasonable physical explanation. But there doesn't exist any such theory at the moment at least, AFAIK.

• No physical theory can be proven in the way mathematical theories can be proven. – Pirx Jan 16 '17 at 0:00
• @Pirx Of course they can be. Navier's stokes existence would be a good example. Ultimately the proof has to be interpreted as something physical, but the approach is pure mathematical. – user2277550 Jan 16 '17 at 0:09
• No. Apart from the fact that the Navier-Stokes problem is open, what you are talking about is a proof of a mathematical statement. That is very different from "proving" a physical statement, which is impossible as I explained above. For the Navier-Stokes example, you would have to provide a proof that these equations describe the motion of certain fluids. No such proof is possible in the same sense mathematical proof is possible. Of course, I am leaving aside the fact that we already know that the Navier-Stokes equations do not constitute an exact description of any physical flow. – Pirx Jan 16 '17 at 0:13
• Yes, my question is more along the lines of how the world itself prevents physics from progressing, or something along those lines. – Simply Beautiful Art Jan 16 '17 at 0:18
• Of course it can't be too exact coz at the atomic level the laws change drastically. But that can be said about any theorem including QM. Your statement that Navier Stokes does not describe any situation is absurd. Into any inquiry in physics you are basically just searching for solutions within certain constraints, and seeing how far a certain mathematical model gets you along that path. But there have been whole areas of mathematics invented for the sake of furthering something in physics. Calabi-Yau, Electroweak. – user2277550 Jan 16 '17 at 0:23

Gödel's theorem requires a written language (for example English), arithmetic including prime numbers, and some axioms from which you try to deduce whether sentences in the language are true or false. Gödel wrote a sentence that could not be proved true of false.

Physics seems to meet Gödel's requirements. We write sentences in a language, use arithmetic with prime numbers, and have axioms from which we deduce sentences about the outcome of measurements as true or false.

At first one might conclude that Physics will never be complete. There will be some sentence we cannot prove from the axioms, yet we can make a measurement with our instruments to determine its truthfulness. Therefore, we have to add yet more axioms to predict the measurement...Physics is never complete.

However, we have already provided in our theories, that some sentences are not answerable from the axioms or by measurement. For example, "The phase of the electron wave function about the hydrogen atom at a particular (x,y,z,t) is $\pi$ radians" true or false? .... a question we acknowledge can't be answered by measurement.

So, perhaps Physics will one day have a complete set of axioms, and the unanswerable Gödel questions will be exactly the ones the theory says measurement can't answer.