Mathematical Logic

Is this a Theory of Everything?

I seem to be describing much of physics, quantum theory, quantum field theory, the particles of the Standard model, a description of the nature of spacetime, and hints of general relativity. So the question arises, is this formulation capable of explaining everything physical in reality? Can this result in a Theory of Everything?

The definition of a theory of everything is that there is no entity or event in nature that is not derivable or describable in terms of the basic axioms of that theory. Since I started with propositional logic and introduced math as needed, I seem to have developed a formal system of axioms. So the question translates into whether this framework composes an axiomatic formal system that is complete and consistent in the mathematical sense? Are there entities in this system that are not derivable in this system? These question are address in the study of mathematical logic.

Mathematical logic is a rather complicated subject, and I'm certainly no expert in it. But one of the most famous results of mathematical logic is called Gödel's Incompleteness Theorem expressed by Kurt Gödel in 1931. It basically states that for any formal system of logic that allows elementary arithmetic on natural numbers, it is possible to construct statements in that system that cannot be proved or refuted by that system. And if physical theories are formal systems of math, then unprovable statements in the math of the theory would correspond to unexplainable events in nature, such as new types of particles or events that seem to defy the theory.

The Incompleteness Theorem has been used by some philosophers to deny the very possibility of a theory of everything. As they say, since any theory of physics is expressed with math, the incompleteness of math means the incompleteness of any theory as well. Some even go further to say that if reality itself is a formal system of some sort, then it too is incomplete, and there can occur events that are beyond the influence of nature. Perhaps, they explain, this is the cause of life, mind, spirit, and other supernatural events.

But strictly speaking, Gödel's Incompleteness Theorem only has to do with formal systems of natural numbers, and other systems that can be mapped to them. Yet, as stated here, "the natural first-order theory of arithmetic of real numbers (with both addition and multiplication), the so-called theory of real closed fields (RCF), is both complete and decidable, as was shown by Tarski (1948)." And it is far from proven that nature is discrete at every level. Even quantum theory gets its quantized values from the solution of differential equations of continuous time and space variables. So I doubt very much that the Incompleteness Theorem applies to theories of physics. Besides, in the present paradigm, the laws of physics are not derived from any formal system of logic; physics is just a matter of using trial and error or intuitive guessing to finding math that fits the measured data of experiment. Why that math and not some other is left unanswered. They use it because it works. So there does not seem to be the ability to even claim it is a formal system of logic subject to the Incompleteness Theorem.

But does the theory I derived here constitute a formal system that is subject to the Incompleteness Theorem? I don't think so. Yes, I started with propositional logic, which is complete. And yes, I arithmetized propositional logic by using the Dirac measure. This allowed me to map disjunction to addition and conjunction to multiplication and implication to the Kronecker delta function. That gives me addition and multiplication on 0 and 1. But can the addition and multiplication used on 0 and 1 be extended to the rest of the natural numbers by mathematical induction? I don't know. Maybe I've gotten Peano arithmetic and don't know it yet. If so, then that much may be subject to Gödel's Incompleteness Theorem.

But I did not stop there. I shrank down the discrete Kronecker delta to the infinitesimal form, the Dirac delta function. This allows me to use all the numbers on the real line, at least from 0 to 1. So it seems I have addition and multiplication on the real line which is complete and decidable according to Tarski (1948). But I also used the complex numbers in the exponential functions to represent the Dirac delta function. And here I'm told that complex numbers are also complete and decidable for the same reason as the real numbers, that "they can't encode and computably deal with finite sequences". I suppose the same holds true for the quaternions and octonions as well.

So I do believe that my theory is complete and decidable, which is one of the requirements of a theory of everything. Another requirement is that it is consistent with every measurement and observation. Of course, it is practically impossible, if not inherently impossible, to measure everything in the finest detail to make sure the theory is consistent with every possible measurement. But if ever an observation contradicts a theory, the theory must be ruled out as a theory of everything. I see no contradictions in my theory yet.

However, Kurt Gödel had to use statements constructed within the formal system that were self-referential, statements that refer to themselves. The Gödel sentence states to the effect that "this statement is not provable", only written with the math characters and rules of that formal system. You can see that this sentence refers to itself and its provability. So even if a formal system developed here is complete and consistent, it's still possible to construct self-referential statements using the syntactical rules of the system.

In mathematical terms self-reference can be expressed by recursion relations (ref1, ref2), which starts with a basic defining function and iterations are defined by inserting the basic function into the function itself. In this way the function then refers to itself by using the previous iteration as an argument in the next iterations, as stated here. The Dirac delta function might be a good example. We start with,

$δ (x_{1} - x_{0}) = \infty$ at $x_{0} = x_{1}$ , and 0 otherwise.

Then,

$δ (x_{2} - x_{0}) = \int_{- \infty}^{\infty} δ (x_{2} - x_{1}) δ (x_{1} - x_{0}) d x_{1}$ .

Then the next recursive iteration will be,

$δ (x_{3} - x_{0}) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} δ (x_{3} - x_{2}) δ (x_{2} - x_{1}) δ (x_{1} - x_{0}) d x_{2} d x_{1}$ .

And the n^th recursive iteration will be,

$δ (x_{n} - x_{0}) = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \cdot \cdot \cdot \int_{- \infty}^{+ \infty} δ (x_{n} - x_{n - 1}) δ (x_{n - 1} - x_{n - 2}) \cdot \cdot \cdot δ (x_{1} - x_{0}) d x_{n - 1} \cdot \cdot \cdot d x_{1}$ .

Notice that $x_{n - 1}$ also takes on the value of $x_{0}$ at some point in the integration process. So the $δ (x_{n} - x_{0})$ being defined is also inside the integrals defining it. Not only this, but inside each Dirac delta function that has both variables being integrated over, there will occur values that make it equal to $δ (x_{n} - x_{0})$ . And the defining function references the function being defined many times. Not only this, but there can be an infinite number of multiplications of the same type of function (the Dirac delta function) being summed over an infinite number of times. This makes this function extremely self-referential.

Yet I used the Dirac delta function to derive the path integral of quantum mechanics by representing the Dirac delta function with a complex Gaussian exponential function, as shown in the previous article, here. The path integral is summing up every possible path from start to finish. So the path integral is referencing every other point, path, and possibility in the integration process. And if there is a quantum mechanical basis to every physical event, then each event that happens in reality takes into account the effects of every other event that happens. Quantum theory has reflectivity and universality properties that make it a self-referential system as defined here.

In math, self-reference is seen in recursion, functions that are defined in terms of more basic functions of the same type being defined. Recursion, then, requires a greater level of complexity, basic functions inside larger functions, in order to be expressed. In nature, then, self-reference should be seen in greater levels of complexity. As more complex structures arise, they depend more sensitively on the exact configuration of their parts, which can be taken as a definition of complexity, as defined here. And structures with less freedom in the number of ways its constituents can be configured have less entropy than those structures whose constituents have a greater number of ways they can be configured. You can see then that complexity and entropy are related.

But in the world we see around us, complexity arises with the passage of time. No complex structures arose at the very instant of the big bang. This means that we must search the math of physics for mechanisms which allow complexity to arise with time. Are there any restrictions on entropy in a given volume of space? Are there any principles that require the axioms of a system to emerge more evidently with the increase of some parameter (time)? If we find such a principle, this might explain the emergence of complexity such as life, and mind, and culture. It might also tell us our destiny and the purpose of life.

If you have any comments, please let me know. Thank you.