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)

Pillars for the THEORY OF EVERYTHING

THE FOLLOWING ARE SOME THEORIES SIMILAR OR CLOSE TO THE ToE.

Kaluza–Klein theory-

Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. It says that the world is 5 dimensional and not 4 dimensional (which is the accepted norm), but we can't see the 5th dimension because it is incredibly small. This extra dimension is packed up tightly into a tiny circle of size in order of Planck length and it would contain just one force and just one type of field, the gravitational force or the gravitational field. The ripples or vibrations produced in this gravitational field would seem to the observers like vibrations in an electromagnetic field and a scalar field in the remaining three spatial and one time dimension. So, it states that all other forces are actually derived from the force of gravity which is itself higher dimensional.

David Hilbert

Hilbert's program-

In mathematics, Hilbert’s program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Basically, it is a theory of everything for mathematics.

The Standard Model-

It is the theory of particle physics concerning the electromagnetic, weak, and strong nuclear interactions which govern the motion and interactions of the sub atomic or elementary particles. The standard model is also referred as the set of all elementary particles known till date (can be considered as a 'periodic table' for particle physics) It includes the force carriers- the gluons, photons, W bosons, Z bosons which are underneath any force or interaction between big and small bodies. The standard model also has the mass carrier- the Higgs boson which are believed to be the fundamental particles that provide mass to any object. The standard model is sometimes referred as "theory of almost everything"due to its inability to incorporate gravity and its foundations being based on simplifying assumptions. Still, it is a quite successful and well tested theory in physics.

Grand Unified theory-

A Grand Unified Theory is a model in particle physics which attempts to merge the electromagnetic, weak, and strong interactions into one single interaction.

Loop Quantum gravity-

Loop quantum gravity is a theory that attempts to describe the 

quantum properties of gravity. It proposes that the structure of space is granular and has a certain critical density of matter that can be contained within it, thus contradicting general relativity (which proposes that space can be stretched infinitely).  Area and volume of any portion of space are quantized.

Granular Structure of Space as proposed by Loop Quantum Gravity

These theories are not ToE but are close to it and will certainly form the FOUNDATIONAL PILLARS of ToE if such a theory would ever be possible to construct.

Click Here for Part-2 (String Theory & E8 Theory)

Click Here for Part-4 (Godel's Theorems)

Two candidates for the ULTIMATE THEORY

String Theory

Projection of a Calabi-Yau Manifold,
one of the ways of compactifying the 
extra dimensions as proposed by 
string theory

String theory is a powerful theory which attempts to solve the ToE. It originated around  1960 and is currently one of the major topics of research in theoretical physics. String theorists believe that in the most fundamental level of matter there is nothing but vibrating strings. These strings are believed to be one dimensional objects present at the heart of all matter and that their vibration produces different particles. This theory proposes that just like musician can produce numerous tunes from a violin string, different types of elementary particles are produced from different vibrations of the strings and these elementary particles form all the matter in the universe. And as there are infinitely many types of vibrations, there are infinitely many types of particles proposed by the string theory. It also predicts that these strings resonate in a mind boggling 10 dimensions. So, as string theory describes the “Standard Model” (the set of all elementary particles observed) as well as naturally incorporates gravity, it is considered as candidate for a ToE by many physicists and string theorists like Michio Kaku, Stephen Hawking and Edward Witten. According to Hawking in particular, "M-theory is the only candidate for a complete theory of the universe."(M-theory is an extension of string theory that predicts existence of 11 dimensions). 

Michio Kaku (leading String Theorist)

The E8 Theory

Antony Garrett Lisi

The E8 theory (as it is popularly known as) is another unified theory which was revealed in November 2007 in a paper known as “An Exceptionally Simple Theory of Everything”. The physicist behind this theory is Antony Garrett Lisi, who is an independent researcher without any academic position. It is based on an 8 dimensional mathematical pattern known as the E8. According to Lisi’s theory of everything, the universe looks like this E8 structure. Lisi says "I think our universe is this beautiful shape." Lisi found a way to put all the elementary particles discovered into the 248 points of the E8 and found 20 points remaining. So, his theory predicts that there must be another 20 elementary particles to complete the standard model. It is basically a geometric unified theory of the universe which effectively combines gravity and particle physics. It is considered to be very promising because it can work within 4 dimensions and does not require any hypothetical higher dimensions as string theory does. Also unlike string theory, the E8 theory is believed to be verifiable by using current technology.  Despite not being a widely accepted theory, this theory has attracted many physicists from round the world who have took it on for further research onto it. But, it is still an incomplete and young theory as described by Lisi and requires time to evolve into a stronger one.

THE E8 PLANE

Click Here for Part-1 (ToE overview)

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

Theory of Everything

The theory of everything (ToE) is the final theory or the master theory that can explain every interaction, every force and the origin of all mass in the universe or simply, a theory to explain everything in the universe as we know it. In other words, it refers to a theory which can explain the origin and existence of the universe and describe its operation in the domain of a single mathematical equation. ToE is one of the major unsolved problems in physics. Not actually an unsolved problem, but an unfulfilled pursuit for the ultimate science. It is currently an active field of research in theoretical physics.

Well, the possibility of discovery of the ToE rests on the two fundamental pillars of modern physics- General Relativity and Quantum Mechanics. Let’s have a brief insight about these two theories.

General Relativity-

It is a theory of gravitation given by Einstein. Modern physics formally came into existence due to the idea of general relativity and therefore Einstein is considered as the father of modern physics. It was indeed a remarkable breakthrough which took physics to a completely different and bizarre path. Well, this theory states that two massive bodies face gravitational attraction towards each other or gravity occurs due to distortions in spacetime caused by massive bodies. It is the current and the most accurate description of gravity in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of spacetime. So, general relativity explains the nature and cause of gravity which is one of the four fundamental forces of nature.

High-precision test of general relativity by theCassini space probe (artist's impression): radiosignals sent between the Earth and the probe (green wave) are delayed by the warping ofspacetime (blue lines) due to the Sun's mass.
(Source-http://en.wikipedia.org/wiki/File:Cassini-science-br.jpg)

Quantum Mechanics-

It is the branch of physics which investigates what happens in the microscopic levels, where the action is on the order of the Planck Constant. In brief, quantum mechanics is the world of the very small. It helps to predict the interactions elementary particles like electrons on the atomic scale. As quantum mechanics deals with tiny particles and interactions at the atomic scale, it explains the three other fundamental forces of nature other than gravity- electromagnetic, weak nuclear and strong nuclear. 

The Standard Model of Elementary Particles

(Source- http://en.wikipedia.org/wiki/File:Standard_Model_of_Elementary_Particles.svg)

The Problem-

Now, for a unified theory of all forces, the four fundamental forces of nature have to be unified into a single force. In other words, we have to unify quantum mechanics and general relativity in order to get a higher theory that can explain all the forces and interactions existing in nature which may prove as the key to the ultimate ToE. But this is where the problem arises. The world of the tiny is not the same as the world of the large. According to quantum mechanics The atomic world is chaotic and nondeterministic. But the world of the large bodies where general relativity comes into play, the situation is drastically different. It is quite deterministic and harmonical with very definite and rigid laws governing their motions. So, the theories of general relativity and quantum mechanics are quite different from each other. And therefore they couldn't have been combined yet. But if there is any hope of finding a ToE, this job has to be done first. Why the quantum world is so different from the world governed by general relativity? Is there any way we can combine both theories to get a satisfying higher theory that is applicable in both scenarios? We can proceed in the hunt for ToE only if these questions are answered. So, rigorous research is currently going on in the field of quantum gravity. Quantum gravity is a field of theoretical physics that seeks to describe the force of gravity according to the principles of quantum mechanics that even gravity can be incorporated into quantum mechanics.

So, if research on quantum gravity can help overcome its limitations and make it a stronger theory, it may be possible that finally we will be able to combine quantum mechanics and general relativity which might give us the key to unlocking the mystery of the universe.

Albert Einstien (Father of Modern Physics)

Max Plank (Father of Quantum Mechanics)

Click Here for Part-2 (String Theory & E8 Theory)