# Can Gödel's Theorem Inform Our Understanding of Physical Reality?

• A
• Giulio Prisco
In summary, Gödel's theorem does not directly imply that no finite mathematical model can capture physical reality. However, it has been theorized by some scientists that there may be a connection between the nondeterminism found in quantum and chaos physics and Gödel's incompleteness. Some scientists, such as Stanley Jaki, Freeman Dyson, and Stephen Hawking, have suggested that this may lead to physical problems that cannot be predicted. However, there is no strong evidence or proof to support this idea. Additionally, changes in the foundations of mathematics, such as abandoning the axiom of choice, are unlikely to have a significant impact on physics. While Gödel's theorem is highly relevant in modern pure mathematics, its practical effects on mathematical physics are limited
Giulio Prisco
TL;DR Summary
What are the implications of Gödel theorem for fundamental science and metaphysics?
What does Gödel’s theorem say about physical reality? Does Gödel’s theorem imply that no finite mathematical model can capture physical reality? Does the nondeterminism found in quantum and chaos physics - it’s impossible to predict (prove) the future from the present and the laws of physics - have something to do with Gödel’s incompleteness?
Many scientists e.g. Stanley Jaki, Freeman Dyson, and recently Stephen Hawking ("According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted…") have formulated this intuition.
But I'm not aware of any proof (or very strong semi-rigorous argument) that causal openness in physical reality follows from Gödel's theorems (or the related results of Turing, Chaitin etc.).

I imagine you have already read a lot on the subject. As far as I can see, there isn't a direct connection from Gödel to physical theories. It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.

It's not a question of predictability, as such. E.g. a physical system that probabilitistically changes (e.g. radioactive decay) may be perfectly simple mathematically. Its unpredictability is not related to the foundations of mathematics, but is simply an axiom of the mathematical model.

With all these questions, I feel you need to be much more specific about what you are saying or asking. What particular aspect of physical theories are affected by the fundamentals of mathematics? E.g. if we abandon the axiom of choice and instead assume that every subset of ##\mathbb R## is Borel-measurable, how would that change in the foundations of mathematics affect physics? If at all?

Jarvis323 and Demystifier
PeroK said:
I imagine you have already read a lot on the subject. As far as I can see, there isn't a direct connection from Gödel to physical theories. It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.

It's not a question of predictability, as such. E.g. a physical system that probabilitistically changes (e.g. radioactive decay) may be perfectly simple mathematically. Its unpredictability is not related to the foundations of mathematics, but is simply an axiom of the mathematical model.

With all these questions, I feel you need to be much more specific about what you are saying or asking. What particular aspect of physical theories are affected by the fundamentals of mathematics? E.g. if we abandon the axiom of choice and instead assume that every subset of ##\mathbb R## is Borel-measurable, how would that change in the foundations of mathematics affect physics? If at all?

Thanks! Let me think about that...

PeroK said:
I imagine you have already read a lot on the subject. As far as I can see, there isn't a direct connection from Gödel to physical theories. It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.
Good point.
PeroK said:
It's not a question of predictability, as such. E.g. a physical system that probabilitistically changes (e.g. radioactive decay) may be perfectly simple mathematically. Its unpredictability is not related to the foundations of mathematics, but is simply an axiom of the mathematical model.

With all these questions, I feel you need to be much more specific about what you are saying or asking. What particular aspect of physical theories are affected by the fundamentals of mathematics? E.g. if we abandon the axiom of choice and instead assume that every subset of ##\mathbb R## is Borel-measurable, how would that change in the foundations of mathematics affect physics? If at all?
As far as I understand, that wouldn't invalidate Gödel theorem, which applies to systems based on weaker sets of axioms than the full ZFC with choice. Please correct me if I am wrong.

In general, I don't think changes in the (or better "a") foundations of mathematics can affect physics.
I tend to think of mathematics as the physics of little things that can be counted - a construction motivated by physical experience and the need to stay alive long enough to reproduce. So thinking beings very different from us might develop entirely different forms of mathematics, and it would be the physics of the real world that discriminates between one or another forms of mathematics. But perhaps Gödel theorem is sufficiently general to apply to all.

Giulio Prisco said:
As far as I understand, that wouldn't invalidate Gödel theorem, which applies to systems based on weaker sets of axioms than the full ZFC with choice. Please correct me if I am wrong.
You can't invalidate Gödel's theorem. I gave an example of the practical effect that Gödel's theorem has on mathematics (it was recommended to me nearly 40 years ago as a potential PhD topic!). To produce a non-measurable set, you must invoke the axiom of choice. And, certainly, measure-theory is central to mathematical physics. Another example is that every Hilbert Space has an orthonormal basis - which in general relies on Zorn's lemma.

The question is whether these undecidable propositions are relevant to physics. In both these cases, the mathematics is probably not relevant - in the sense that any subset of ##\mathbb R## that is used in physics may be assumed to be measurable and every separable Hilbert space has a basis without invoking the axiom of choice.

From that point of view it appears that the subset of mathematics that is used in mathematical physics is unaffected by undecidable propositions - although I'm not claiming that categorically!

@Giulio Prisco by contrast, Gödel's theorem and the axiom choice are highly relevant for modern pure mathematics. It's a part of everyday life!

For example, take a look at the first problem from our monthly maths challenge from last year:

I had a go at this problem and began to suspect that it was undecidable and depended on the axiom of choice. And, indeed, that turned out to be true, via the Baire Category Theorem:

https://en.wikipedia.org/wiki/Baire_category_theorem

Does this sort of thing happen in mathematical physics?

PeroK and Jarvis323
PeroK said:
It's entirely possible, for example, that ultimately the universe has a finite number of possible states, which would make Gödel's theorems irrelevant.
Yes. A lot of trouble in pure math arises from the axiom that an infinite set exists.

Fractal matter and Fra
PeroK said:
although I'm not claiming that categorically!
I'm glad you don't. I can follow arguments based on logic and set theory, but when someone uses category theory I get lost.

Demystifier said:
Yes. A lot of trouble in pure math arises from the axiom that an infinite set exists.
Is it really an axiom?
Just start counting: ##1,2,3,\ldots## it never stops.

MathematicalPhysicist said:
Just start counting: ##1,2,3,\ldots## it never stops.
You just stopped, at 3 followed by the dots.

hutchphd
Demystifier said:
You just stopped, at 3 followed by the dots.
the dots mean that the counting never stops.
We all know there's no last number in the naturals.

MathematicalPhysicist said:
We all know there's no last number in the naturals.
Yes, but the full set of natural numbers is our mental invention. In nature we only see finite subsets of it. Observed physics could, in principle, be formulated with a set in which the largest number is, say, ##100^{100^{100}}##. Or if it's not big enough, then the busy beaver of that would certainly suffice. https://en.wikipedia.org/wiki/Busy_beaver

xristy and Fra
Demystifier said:
Yes, but the full set of natural numbers is our mental invention. In nature we only see finite subsets of it. Observed physics could, in principle, be formulated with a set in which the largest number is, say, ##100^{100^{100}}##. Or if it's not big enough, then the busy beaver of that would certainly suffice. https://en.wikipedia.org/wiki/Busy_beaver
You can only grasp a finite portion of the universe. Beyond the horizon nothing is known.

MathematicalPhysicist said:
You can only grasp a finite portion of the universe. Beyond the horizon nothing is known.
Even if there was no horizon, we could only observe a finite amount of data. The size of the universe has nothing to do with that.

xristy and Fra
Demystifier said:
Yes, but the full set of natural numbers is our mental invention. In nature we only see finite subsets of it. Observed physics could, in principle, be formulated with a set in which the largest number is, say, ##100^{100^{100}}##. Or if it's not big enough, then the busy beaver of that would certainly suffice. https://en.wikipedia.org/wiki/Busy_beaver
One question one could ask from an agent interpretation is; how far can an agent count? or, what observational resolution does it have? Does this have any relations to the agents internal coding? mass? And if this is so, would this natural regulator, have any impact on the agent-agent interactions?

All these interesting questions are LOST at the starting point, as you pull the real number out of the sleeve. It is also a problem already in Jayens and others reconstruction of probability theory. For this reason even starting from pure probilistic reasoing is not innocent.

Edit: Note that this objection is not suggesting that the universe has to be finite, just that any intrinsic perspective should be limited. We have to be able to measure this. And if the possibilities are always infinte, we face situations where infinity fights infinity, and a post-construction of regularisation of formally non-sensiable expressions is required from us. I think we can do better.

/Fredrik

Delta2
Fra said:
One question one could ask from an agent interpretation is; how far can an agent count? or, what observational resolution does it have? Does this have any relations to the agents internal coding? mass? And if this is so, would this natural regulator, have any impact on the agent-agent interactions?

All these interesting questions are LOST at the starting point, as you pull the real number out of the sleeve. It is also a problem already in Jayens and others reconstruction of probability theory. For this reason even starting from pure probilistic reasoing is not innocent.

Edit: Note that this objection is not suggesting that the universe has to be finite, just that any intrinsic perspective should be limited. We have to be able to measure this. And if the possibilities are always infinte, we face situations where infinity fights infinity, and a post-construction of regularisation of formally non-sensiable expressions is required from us. I think we can do better.

/Fredrik
I must confess I have no idea what any of that means - especially the last paragraph.

PeroK said:
I must confess I have no idea what any of that means - especially the last paragraph.
I see that I brought in too much thinking at once without giving it the time needed to formulate it. For clarity I might as well have deleted that last paragraph. The two first ones, was the main points anyway.

(What I associated to in that paragraph was relations between the cardinality or the observable event spaces and renormalisation problems in general, as we may miss out "natural regulators" which has a physical interpretation. The result can be formal expressions that are divergent, but which shouldn't had to be, if only the physical complexions was counted to start with)

/Fredrik

Fra said:
I see that I brought in too much thinking at once without giving it the time needed to formulate it. For clarity I might as well have deleted that last paragraph. The two first ones, was the main points anyway.

(What I associated to in that paragraph was relations between the cardinality or the observable event spaces and renormalisation problems in general, as we may miss out "natural regulators" which has a physical interpretation. The result can be formal expressions that are divergent, but which shouldn't had to be, if only the physical complexions was counted to start with)

/Fredrik
That's not any better!

Giulio Prisco said:
Summary:: What are the implications of Gödel theorem for fundamental science and metaphysics?

What does Gödel’s theorem say about physical reality? Does Gödel’s theorem imply that no finite mathematical model can capture physical reality? Does the nondeterminism found in quantum and chaos physics - it’s impossible to predict (prove) the future from the present and the laws of physics - have something to do with Gödel’s incompleteness?
Many scientists e.g. Stanley Jaki, Freeman Dyson, and recently Stephen Hawking ("According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted…") have formulated this intuition.
But I'm not aware of any proof (or very strong semi-rigorous argument) that causal openness in physical reality follows from Gödel's theorems (or the related results of Turing, Chaitin etc.).
"The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers."
"For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system."

Basically, the 1st Gödel’s theorem is about natural numbers. The set on axioms should contain natural numbers in it for the Gödel’s theorem to apply. In the opposite case the Gödel’s theorem doesn't apply. Gödel’s theorem states that if you take a set on axioms that contain natural numbers there are statements, that are undecidable(neither true not false). Those can not be proved to be true or false, considering that set of axoims, and need to more powerful set of axioms to be proven. The cause is the infinity as i understand it.
Demystifier said:
Yes. A lot of trouble in pure math arises from the axiom that an infinite set exists.

Demystifier

## 1. What is the significance of Gödel's Incompleteness Theorems in relation to physical reality?

The Incompleteness Theorems, first proposed by mathematician Kurt Gödel, prove that in any formal system of mathematics, there will always be true statements that cannot be proven within that system. This has implications for our understanding of physical reality, as it suggests that there may be fundamental truths about the universe that cannot be fully understood or proven through mathematical or scientific means.

## 2. Can Gödel's Incompleteness Theorems be applied to physics?

Yes, the Incompleteness Theorems have been applied to various branches of physics, including quantum mechanics and cosmology. Some scientists have even proposed that the universe itself may be a type of formal system, subject to the limitations of Gödel's theorems.

## 3. How does Gödel's Incompleteness Theorems challenge our understanding of causality?

Gödel's theorems suggest that there are truths about the universe that cannot be proven or explained through cause and effect relationships. This challenges the traditional scientific approach of seeking to understand the universe through causal explanations and highlights the limitations of such an approach.

## 4. Is there a connection between Gödel's Incompleteness Theorems and the concept of free will?

Some philosophers and scientists have proposed that Gödel's theorems have implications for the existence of free will. If there are truths about the universe that cannot be proven or predicted, then it may suggest that there is room for free will to exist outside of deterministic laws.

## 5. How have Gödel's Incompleteness Theorems influenced the philosophy of science?

Gödel's theorems have sparked debates and discussions about the limitations of scientific knowledge and the boundaries of human understanding. They have also led to the development of new theories and approaches in fields such as quantum physics, where the concept of uncertainty and indeterminacy align with the ideas put forth by Gödel.

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