Does the first Godel's incompleteness theorem forbids the existence of a Theory of Everything?

Question

There is some position that the first Godel's incompleteness theorem does refute the possibility of finding a Theory of Everything.

But the first Godel's incompleteness theorem stands:

In every formal system with capacity to express Peano Arithmetric is either incomplete or inconsistent.

It is clear that our universe is capable of express Peano arithmetic, but if the system had a unreachable-transfinite number of axioms [1], then there could be a consistent and complete fundamental set of axioms ruling the physical phenomena.

So in summary, is it true that the first Godel's incompleteness theorem implies that there is no Theory of everything?

[1] https://math.stackexchange.com/questions/309147/system-with-infinite-number-of-axioms.

Edit: I would like to explain that, here I am refering to a TOE in the sense of a theory that explains all the physical phenomena.

Answer 1

Godel's theorem applies to systems of formal logic. A Theory Of Everything (TOE) is a specific term used to describe the union of quantum mechanics (QM) and general relativity )GR). Note that this union would not be a mathematical model of every phenomenon in the universe in that it would not allow things like viscoelastic behavior, non-newtonian fluid flow or trans-sonic aerodynamics to be directly derived from its first principles.

As such, Godel's incompleteness theorem will not prevent QM and GR from being unified into one TOE... but at present, no one knows how to accomplish this.

Answer 2

"Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end" -Stephen Hawking in his lecture Godel And The End Of Physics.

The Penrose-Lucas argument goes

"while a formal proof system cannot prove its own consistency, Gödel-unprovable results are provable by human mathematicians. He takes this disparity to mean that human mathematicians are not describable as formal proof systems, and are therefore running a non-computable algorithm"

There certainly are people who disagree with this, but I think the ideas of a Nobel Prize winner with many substantial contributions to physics, should at least not be dismissed out of hand (though I fully expect other answers or comments to do so).

A unified field theory, or quantum-gravity, no problem. But a Theory Of Everything is intrinsically a goal arising from hubris (like the Hilbert programme, that Godel ended). It's not even clear precisely what it means, how it could ever be discrete or finished, it would have to include every aspect of knowledge and experience, with some degree of predictions.

But can we fully predict conscious beings, without making models that are, conscious? When a theory has conscious elements, it's a mind, or contains them.

I see Hofstadter's strange loops and tangled hierarchies, as the way out of Godel Incompleteness, by sidestepping the foundations for knowledge of the Munchausen trilemma: axioms cannot justify axioms, circular reasoning cannot justify circular reasoning (eg problem of induction), infinite regress is possible but must exceed a finite system of knowing, you could never verify it. So we cannot prove anything in a foundational way - only within a system. Strange loops point to how consciousness can arise within a system, as strange loop feedback, and through a process of tangling hierarchies build a process of coherentism, in which strange loops develop emergent capacities to condition the system rather than be conditioned by it. This has been identified as a repeated motif in philosophy, eg Wittgenstein's ladder. In this view proof systems are not out there, but within us, specifically within the collaborative intelligence of language. And with new words, new ideas, that changes, and new truths are possible.

A mind can begin where it is, and 'on the fly' generate a new system, in a way formal systems cannot. This is what avoids the halting problem and creates Godel incompleteness: a mesh of interacting priorities that can seek to highlight and solve problems with reference to an entire tangled hierarchy, inc adding layers. The truth unprovable in the previous system, is recognised as truth by a mind because it has changed the system (typically adding axioms).

So, a complete system is possible, but would involve creating a 'theory' that could continue to elaborate itself, ie a mind capable of thought. I'd link this to the emergent mutually-arising collaborative network of Indra's Net, rather than any prexisting monistic mind with foreknowledge. The universe coming to understand it's own self, not as a conditioned organism, but as a choice how to be.

Answer 3

First of all is important to distingish between the idea that the universe is governed by a fundamental set of axioms, and that we can have a Theory of Everything.

Second the first Godel's incompleteness theorem, clearly implies that there is no fundamental set of axioms that could rule the universe, that is both complete and consistent, ergo the real implication of the Godel's theorem would be that can be situations where the fundamental axioms that rule the physical phenomena would be unable to dictate a clear outcome. But does no speak about our ability to unify our current knowledge of the physical phenomena.

Hence the the Idea that the 1st Godel's incompleteness theorem no disproves the possible existence of a TOE (defining TOE as the unification of our current knowledge of the physical phenomena), is false. And I argue that is methaphisically nesesary to exists a TOE, because if we cannot unify our current the it likely would mean that the universe is inconsistent.