GöDEL's THEOREMS- "the argument against Theory of Everything"

Gödel's incompleteness theorems are two theorems of mathematical logic that establish limitations against completeness or consistency of any physical or arithmetic theory. They were devised by Kurt Gödel in 1931.

Kurt Gödel 

1st INCOMPLETENESS THEOREM-

“For any consistent axiomatic system in arithmetic there always exists a set of arithmetic propositions which can’t be proved by it.”

It means that, a consistent (free from any sort of contradiction) mathematical theory-X that completely covers up certain mathematical topic-x cannot be stated as a complete theory because it is always possible to formulate a statement belonging to topic-X whose validity or invalidity cannot be proved by theory-X.

So, basically, Gödel’s first theorem states that a theory can’t be consistent as well as complete at the same time and therefore there would always exist unprovable statements.

2nd INCOMPLETENESS THEOREM-

“Any consistent axiomatic system cannot prove its own consistency and if it does so, it is inconsistent.”

So, the 2nd theorem clearly states that any consistent mathematical theory cannot be used to prove itself.

Now, what has Gödel’s theorems got to do with Theory of Everything?

First we must remember that a physical theory is always in form of a mathematical model. So, the Gödel’s theorems hold as good for physics as they do for mathematics. Now, suppose we get a theory that could be claimed as a theory of everything (call it ToE v1.0). For it to be accepted as valid, it is bound to be consistent. So, as it is consistent, by Gödel’s 1st theorem, it will be incomplete or there will always be something that couldn't be explained by it (thus contradicting it from a theory of “EVERYTHING”). Now, even if we construct a even higher theory (call it ToE v2.0) which could explain the thing that was unexplainable by ToE v1.0, Gödel’s 1st theorem would still apply, due to which something else will get formulated which would be unexplainable by ToE v2.0. So, this process of refinement of theories will go on forever but still none of them will succeed to be claimed as a true Theory of Everything.

Suppose we are able to create a satisfying theory of everything disregarding Gödel’s first theorem. Now, this theory by definition would be able to explain everything. And everything includes itself. This is where Gödel’s 2nd theorem will come to play. By Gödel’s 2nd theorem, we know that no theory can prove itself. So, if this theory of everything could explain itself, it will be inconsistent and eventually be invalid.

Therefore Gödel’s incompleteness theorems will deviate any proposed theory of everything from its very definition of being “of EVERYTHING” and will disprove its validity. So, this is how Gödel’s theorems prove to be strong arguments against a theory of everything.

A number of scholars claim that Gödel's incompleteness theorem suggests that any attempt to construct a ToE is bound to fail. Stephen Hawking was originally a believer in the Theory of Everything but, after considering Gödel's Theorem, concluded that one was not obtainable. 

He says In his lecture ‘Gödel and the End of Physics’ -

“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, and that we will always have the challenge of new discovery. Without it, we would stagnate. Godel’s theorem ensured there would always be a job for mathematicians. I think M theory will do the same for physicists.”

Still many physicists believe that a ToE is possible to formulate. Most physicists believe that Gödel's Theorem does not mean that a ToE cannot exist. They argue that the theory of everything may not refer to a set of underlying rules but to the understanding of how and why the universe exists and how it works. Surely they will leave no stone unturned to get to it.

(To tell the truth I as a lover of physics am quite scared and startled of the consequences of these theorems)

Click Here for Part-3 (Pillars for the Toe)

Click Here for Part-5 (Reasons and Consequences of the ToE)