Do Gödel’s Incompleteness Theorems Exclude the Possibility of a Theory of Everything?

Many physicists, Prof. Stephen Hawking being the most recognizable, interpret Gödel’s incompleteness theorems as forbidding even the possibility a theory of everything (read Prof. Hawkins thoughts here).

Prof. Hawking writes:

“What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. 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.”

But mathematics and physics are not comparable. There are no restrictions on the formulation of a mathematical proposition so it is always possible to formulate a proposition that cannot be derived from a given axiom set or shown that it is inconsistent with it. If proposition cannot be derived from the axiom set, then the axiom set is considered incomplete. Additionally, if the proposition is inconsistent with the axiom set, then any axioms that are added that would make it possible to derive the proposition would be internally inconsistent. Therefore, all axiom sets are either incomplete, inconsistent or both.

Things are very different in physics. We can safely assume that the Universe is composed of a finite number of types of fundamental objects which interact through a finite number of fundamental forces and that together they define a finite number of fundamental laws; all of which may be represented by an axiom. From the set of all such axioms, the axiom set of the universe, can be derived the behaviour of any physical system at all scales. It is not possible to derive a prediction from this axiom set that would be inconsistent with it or that would not be part of universe at some point of its evolution. The Universe’s axiom set being complete (all that is can be derived from its axiom set) and consistent (nothing that is derived from its axiom set is inconsistent with it), then the theory derived on the Universe’s axiom set must be the theory of everything. The theory of everything is therefore possible.

But though a theory of everything is possible, how can we find it and how can we know that it is when we do?

A physical prediction can be tested without knowing the Universe’s axiom set. We don’t have to reduce a prediction to the axioms of axiom set of Universe to prove it. We don’t need to know the axiom set. All we have to do is devise observations or experiments that test it. If the prediction is not consistent with observations, and if the prediction is correctly derived from a physics theory, then we know that the axiom set of the theory is incomplete and/or inconsistent. If all predictions at all scales are observationally confirmed, then the theory is complete and consistent and we have a theory of everything.

Quantum mechanics and general relativity both make predictions that are consistent with reality at a microscopic and macroscopic scales respectively, but when applied to all scales both theories make predictions that are inconsistent with observations or experiments. This proves that both theories are incomplete and/or inconsistent or that their axioms do not correspond to fundamental aspects of reality. It also proves that their unification is impossible since their axiom sets are mutually exclusive. If we are to make progress towards the theory of everything (I use “the” because there could only be one), then we will have to work from axiom sets different than those of general relativity and quantum mechanics. A good place to start may be minimal axiom sets necessary to describe dynamic systems.

Note: the following is taken from Introduction to Quantum-Geometry Dynamics.

Two Ways to do Science

From an axiomatic standpoint, there are two only two ways to do theoretical physics. The first aims to extend, expand and deepen an existing theory; which is what the overwhelming majority of theorists do. This approach assumes that the theory is fundamentally correct, that is, its axioms are thought to correspond to fundamental aspects of reality.

The second way of doing theoretical physics is to create a new axiom set and derive a theory from it. Distinct axiom sets will lead to distinct theories which, even if they are mutually exclusive may still describe and explain phenomena in ways that are consistent with observations. There can be a multiplicity of such “correct” theories if the axioms are made to correspond to observed aspects of physical reality that are not fundamental but emerging. For instance, theories have been built where one axiom states that the fundamental component of matter is the atom. Such theories, though it may describe very well some phenomena at the molecular scale will fail in explaining a number of phenomenon at smaller scales. In the strict sense, premises based on emergent aspect of reality are not axioms in the physical sense. They can better be understood as theorems. And as mathematical theorems in mathematics can explain the behavior of mathematical objects belonging to a certain class but cannot be generalized to others, physical theorems can explain the behavior of class of objects belonging to a certain scale but these explanations cannot be extended to others scales or even to objects or other classes of objects in the same scale.

But axiom sets are not inherently wrong or right. By definition, since axioms are the starting point, they cannot be reduced or broken down. Hence, as such, we cannot directly prove whether they correspond to fundamental aspects of reality. However, if the models that emerge from an axiom set explain and describe reality and, most importantly, allows predictions that can be tested, then confirmation of the predictions become evidence supporting the axiom set.

The Axiomatic Approach

It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.     Albert Einstei

The dominant approach in science (and a hugely successful one for that matter) is the empirical approach. That is, the approach by which science accumulates data from which it extracts relationships and assumptions that better our understanding of the Universe.

The empirical approach is an essential part of what one which we might call deconstructive. By that I mean that we take pieces or segments of reality from which, through experiments and observations, we extract data from which we hope to deduce the governing laws of the Universe. But though the deconstructive approach works well with observable phenomena, it has so far failed to provide us with a complete and consistent understanding of fundamental reality.

Of course, when a theory is formulated that is in agreement with a data set, it must be tested against future data sets for which it makes predictions. And if the data disagrees with predictions, the theory may be adjusted so as to make it consistent with the data. Then the theory is tested against a new or expanded data set to see if it holds. If it doesn’t, the trial and error process may be repeated so as to make the theory applicable to an increasingly wider domain of reality.

The amount of data accumulated from experiments and observations is astronomical, but we have yet to find the key to decipher it and unlock the fundamental laws governing the Universe.

Also, data is subject to countless interpretations and the number of mutually exclusive models and theories increases as a function of the quantity of accumulated data.

About the Source of Incompatibilities between Theories

Reality can be thought as an axiomatic system in which fundamental aspects correspond to axioms and non-fundamental aspects correspond to theorems.

The empirical method is essentially a method by which we try to deduce the axiom set of reality, the fundamental components and forces, from theorems (non-fundamental interactions). There lies the problem. Even though reality is a complete and consistent system, the laws extracted from observations at different scales of reality and which form the basis of physics theories do not together form a complete and consistent axiomatic system.

The predictions of current theories may agree with observations at the scale from which their premises were extracted, but they fail, often catastrophically, when it comes to making predictions at different scales of reality.

This may indicate that current theories are not axiomatic; they are not based on true physical axioms, that is; the founding propositions of the theories do not correspond to fundamental aspects of reality. If they were, then the axioms from distinct theories could be merged into a consistent (but not necessarily complete) axiomatic set. There would be no incompatibilities.

Also, if theories were axiomatic systems in the way we described above, their axioms would be similar or complementary. Physical axioms can never be in contradiction.

This raises important questions in regards to the empirical method and its potential to extract physical axioms from the theorems it deduces from observations. The fact that even theories which are derived from observations of phenomena at the microscopic scale have failed to produce physical axioms (if they had, they would explain interactions at larger scales as well) suggests that there is an distinction between the microscopic scale, which is so relative to our scale, and the fundamental scale which may be any order of magnitude smaller.

There is nothing that allows us to infer that the microscopic scale is the fundamental scale or that what we observe at the microscopic scale is fundamental. It may very well be that everything we hold as fundamental, the particles, the forces, etc., are not.

Also, theories founded on theorems related to different scales rather than axioms cannot be unified. It follows that the grand unification of the reigning theories which has been the dream of generation of physicists is mathematically impossible. A theory of everything cannot result from the unification of the standard model and relativity, for instance, them being based on mutually exclusive axiom sets. This is why it was so essential to rigorously derive quantum-geometry dynamics (QGD) from its initial axiom set and avoid at all times the temptation of contriving it into agreeing with other theories.

So even though, as we will see later, Newton’s law of universal gravity, the laws of motion, the universality of free fall and the relation between matter and energy have all been derived from QGD’s axiom set, deriving them was never the goal when the axiomatic set was chosen. These laws just followed naturally from QGD’s axiom set.

However, an axiomatic approach as we have described poses two important obstacles.

The first is choosing a set of axioms where each axiom corresponds to a fundamental aspect of reality if fundamental reality is inaccessible thus immeasurable.

The second obstacle is how to test the predictions of an axiomatically derived theory when the scale of fundamental reality makes its immeasurable.

In the following chapters, we will see that even in the likely scenario that fundamental reality is unobservable, if the axioms of our chosen set correspond to fundamental aspects of reality then there must be inevitable and observable consequences at larger scales which will allow us to derive unique testable predictions. We will show that it possible to choose a complete and consistent set of axioms, that is one from which interactions at all scales of reality can be reduced to. In other words, even if the fundamental scale of reality remains unobservable, an axiomatic theory would make precise predictions at scales that are.

Internal Consistency and Validity of a Theory

Any theory that is rigorously developed from a given consistent set of axioms will itself be internally consistent. That said, since any number of such axiom set can be constructed, an equal number of theories can derived that will be internally consistent. To be a valid axiomatic physics theory, it must answer positively to the following questions.

1. Do its axioms form an internally consistent set?
2. Is the theory rigorously derived from the axiom set?
3. Are all descriptions derived from the theory consistent with observations?
4. Can we derive explanations from the axiom set that are consistent with observations?
5. Can we derive from the axiom set unique and testable predictions?

And if an axiom set is consistent and complete, then:

1. Does the theory derived from the axiom set describe physical reality at all scales?

In the following chapters, we will see how quantum-geometry dynamics answers these questions.