Thursday, July 7, 2022




© Copyright July 2022, Edward R Close

“As a man who has devoted his whole life to the most clearheaded science, to the study of matter, I can tell you as a result of my research about the atoms this much: There is no matter as such! All matter originates and exists only by virtue of a force which brings the particles of an atom to vibration and holds this most minute solar system of the atom together. … We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter. … I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates (the existence of) consciousness.” - Max Planck, the First Quantum Physicist

I completely agree with these statements by Max Planck. My agreement is based on more than sixty-four years of my own on-going study and research that has convinced me that, far from reflecting outdated unscientific religious faith, as many mainstream materialists like to think, Planck’s conclusions gathered above were, as he implies, based on a clearheaded appraisal of scientific investigation based on meticulous experimental design and real data. He was not expressing a wishful fantasy or subjective opinion. He was articulating a set of valid logical conclusions and predicting the science of the future. In my opinion, the conclusions clearly articulated above were written by a genius who was far ahead of his time!

It is, however, easy to say that we, and all of the details of the reality we experience, are materialized thoughts of a higher form of intelligence that is beyond our ability as human beings to comprehend. That appears to be the basis of traditional philosophies of religion and spiritual mysticism. But if Max Planck’s scientific view of reality is correct, and everything is materialized thoughts of an omnipotent transcendental mind, then exactly how does that work? Is it possible that we may be able to understand the processes and mechanisms by which an infinite mind is thinking us into existence? If this is the case, then discovering and understanding those mechanisms and processes will be of immense importance to us as individuals, and even more vital to the future of humanity.

The Way Forward

The last statement of the last discussion, posted July 2nd, listed the three types of potentially consistent logical systems that make up the reality we experience. They are: 1) Incomplete parts of reality that are perceived by individual conscious beings primarily through the senses, 2) Mathematical models of the logical systems of the laws that govern reality, and 3) The finite objective universe visibly expanding into infinity.

The full statement, which is consistent with Planck’s statements, also says that those three logical systems, which are by definition, subject to the conclusions of Gödel’s incompleteness theorem, reflect the innate patterns of logic existing in Primary Consciousness, i.e., the mind of God. But is this merging of science with spirituality actually warranted by empirical evidence? And is there a paradox? Yes, and yes, of course there is, and we shall see that, instead of being a problem, that is a profound blessing and an opportunity to learn!

The Mind of God Paradox

If the three logical systems that form the reality we experience are incomplete, and they are reflections of the logical patterns of the mind of God, then the proof of Gödel’s incompleteness theorem implies that the mind of God is incomplete! This conflicts with the traditional religious belief systems that proclaim the absolute perfection of God. And this conclusion that the Mind of God is incomplete could even be considered as evidence that, instead of being created by a higher form of intelligence, we are, in fact, only creating the concept of a God-like potential ourselves, as we try to expand our collective consciousness by accumulating more and more factual knowledge and understanding more about the nature of reality.

Now, what we have here is what I like to call a Niels Bohr moment! Recall that Bohr said:

How wonderful that we have met with a paradox. Now we have some hope of making progress!”

Here’s the crux of our paradox: How can a Perfect, Infinite God have an incomplete mind?

Let’s remind ourselves what the incompleteness theorem says:

In any consistent logical system, contradictions may arise that cannot be resolved using the logic of the system within which they have been stated.

Or as Einstein put it: “We cannot solve our problems with the same thinking we used to create them.”

It is very important to realize that the incompleteness theorem proved by Kurt Gödel does NOT say that there are paradoxes that can never be resolved. It simply says that there are perfectly valid questions that can be asked in any consistent logical system (modern science, for example) that cannot be answered within the system of logic in which it was articulated.

As I have pointed out before, reality is the ultimate logical system, and paradoxes arising in a consistent system of logic are evidence of either an incorrect or incomplete set of a priori (self-evident) assumptions. Paradoxes can actually be resolved by making appropriate changes in the system of logic we are using. How do we do that? We have to start by critically examining the relevant existing a priori assumptions of our system of logic. What are the a priori assumptions in this case? They are assumptions about the existence and nature of time and infinity.

So, let’s first have a look at the assumptions underlying the concept of infinity, The average person rarely thinks much about infinity, and those who do, including most mainstream scientists, assume that infinity is unattainable in the real world and therefore impossible to define, study, or further analyze. In the late 1800s, however, one mathematician in Germany disagreed with this assumption. That mathematician’s name was Georg Cantor.

Set Theory and Cantor’s Infinity of Infinities

The Russian-born German mathematician, Georg Cantor (1845 -1918), like Max Planck, was a genius 100 years before his time. He boldly investigated the concept of infinity, even though most of the mainstream scientists and mathematicians of his time considered infinity to be a concept of religious philosophy, and therefore, not at all a proper thing for a mathematician to study or even to consider as a real thing. Cantor paid a terrible price for his heresy against the mainstream establishment. After his brilliant discoveries were rejected out-of-hand by a leading member of his chosen profession, he suffered a series of nervous breakdowns and died in a psychiatric sanitorium in Halle Germany at the age of 73.


About 20 years after Cantor died, the methods that he developed to study infinities were generally accepted. Interestingly, they are the exact same methods used by Kurt Gödel to prove the incompleteness theorems. What are those methods? They are the fundamental methods of what is now known as Set Theory, a theory of mathematical logic that has proved to be very useful in many fields of scientific investigation. The concept of “a set” is so simple that it is often introduced in basic math classes as if it were a completely self-evident concept. Examples of sets are everywhere: like the set of desks in a classroom, the set of students in a class, the set of cars in a parking lot, etc. The concept of set is strongly related to the fundamental functioning of consciousness. So how does set theory handle the concept of infinity?


From a purely mathematical point of view, there are an infinite number of infinities, and every infinite set is based on one or more provable axioms or a priori (self-evident) assumptions. The idea that there could be different sizes of infinity seems counter-intuitive, but an example of a set of infinities of different sizes that can be easily visualized is the set that includes the infinite set of points on a line, the infinite set of points in a plane, and the infinite set of points in a volume of space. These form a sequential set of infinite sets where the first set is a subset of the second set, and both are contained in the third set. The number of points on a line is infinite, but since a plane contains an infinite number of lines, the infinity of points in a plane is larger than the infinity of points on a line, and the infinite set of points in a volume is even larger. While this seems intuitively obvious to me, a rigorous proof that there are infinite sets of at least one size between the size of countable sets and the size of uncountable infinite sets depends upon a famous proposition, the continuum hypothesis, being false. FYI, Cantor postulated that the continuum hypothesis is true, but failed to be able to prove it during his lifetime.


The continuum hypothesis says that there are two different sizes of infinite sets. The smaller-sized infinite sets have a one-to-one correspondence with the members of the infinite set of integers (1, 2, 3, …). Cantor called such a set a countable set, for obvious reasons, and the larger size of infinite sets is represented by the infinite set of the real numbers (the integers plus the rational numbers existing between them), which Cantor called uncountable because they could not be paired with the infinite set of integers. The importance of proving whether or not the continuity hypothesis is true, was emphasized by David Hilbert when he made it number one in his famous list of 23 important unsolved problems of mathematics, published in 1900. Only eight of the 23 problems have been resolved to date according to the consensus of professional mathematicians, and the continuum hypothesis is not among the eight that have been resolved.


There are ten basic axioms that are typically introduced by a mathematician in a formal presentations of set theory. [An axiom is defined as a statement that is either assumed as a priori (i.e., self-evident), or has been proved with rigorous logic from an a priori assumption, and is therefore accepted as true.] There are potentially an infinite number of set theory axioms because some axioms are themselves infinite sets. However, in our effort to resolve the Mind of God paradox, we only need to undertake a critical examination of the a priori assumptions of set theory that are directly related to the fundamental concept of the existence of infinite sets. Here is a set of relevant axioms and axiomatic definitions:

·        The axiomatic definition of set theory: Set theory is the system of mathematical logic that deals with collections of similar objects called sets. The objects that make up the sets are called members, or elements, of the set. Pure set theory is a calculus that deals exclusively with sets, so the only sets under consideration are those whose members are also sets. The theory of the hereditarily finite sets, i.e., those finite sets whose elements are also finite sets, the elements of which are also finite, is actually formally equivalent to the calculus we call arithmetic. Since arithmetic is already well-defined and axiomatically sound, the main focus and purpose of set theory is the study of infinite sets.

·        The axiom of the existence of infinity: There exists an infinite set that contains all definable sets (both finite and infinite) as elements or subsets, including the zero, or empty set.

·        The axiomatic definition of countable infinities: An infinite set is countable, if and only if all of its elements can be paired with the infinite set of integers (1, 2, 3,…).

·        The Axiomatic definition of uncountable infinities: An infinite set is uncountable if its elements cannot be paired with the infinite set of integers.

·        The axiomatic definition of equally infinite sets: Two infinite sets are the same size, if the elements of one set can be paired one-to-one with the elements of the other set.

·        The axiom of sizes of infinities: There are at least two different sizes of infinities: The size of countable infinite sets, which is equal to the size of the infinite set of integers and the size of uncountable infinite sets like the infinite set of real numbers, which contains the infinite set of integers and the rational numbers existing between them, making uncountable infinite sets necessarily much larger than countable infinite sets.



The fact that the dynamic interactions of the existing logical systems of the three types listed in the previous post and again above, have produced the stable reality that we experience, provides us, as conscious beings, with a unique opportunity to investigate and understand the nature of reality. In the process of doing so, we came upon a striking paradox that I have called “The Mind of God paradox”.

Combinations of the three types of logical systems cannot form a physical universe stable enough to support conscious life forms, unless they conform to the same symmetric patterns of rotational volumetric extent that exists in the innate logical dimensional structure of space, time, and consciousness. The symmetry needed for the stability we experience in our natural environment is provided at the quantum level by specific numbers of triadic rotational units of equivalence (TRUE) of gimmel, the non-material third form of objective reality, discovered while applying the calculus of dimensional distinctions to analyze the stability of the combination of two up quarks and one down quark in the formation of a proton. And, of course, as we all know, each positively charged proton pairs with a negatively charged electron to form hydrogen, by far the most abundant element in the universe.

What does hydrogen have to do with set theory, the incompleteness theorem, space, time, consciousness, and the mind of God paradox? I’m glad you asked! The hydrogen atom is a finite set with three finite subsets: one electron energy shell, enclosing one proton, containing three quarks, comprised of an asymmetric number of quantal units (TRUE) of mass and energy, which is a rotational set of objects that only becomes stable in union with an appropriate number of units of gimmel. Gimmel, like the dimensions of space, time, and consciousness, has no physical existence, i.e., no mass or energy, of its own, but the existence of gimmel in the electron and quarks, expands the hydrogen atom into a symmetrical rotating structure that would be unstable without it, in which case, there would be no universe to support organic life, the vehicle of conscious minds, a finite set within the infinite set of logical systems that is the Primary form of Consciousness called the Mind of God.


Our study of set theory axioms related to infinity has revealed an error in the assumptions that gave rise to the Mind of God paradox. That error is the mistake of equating perfection with completeness. The exact opposite it true: The Primary form of Consciousness, i.e., the Mind of God, is infinite, but we didn’t realize that infinity is never complete. This conclusion is consistent with the truth revealed in the proof of the incompleteness theorem. Expansion of the consciousness of the finite is movement toward perfection, but perfection is a moving target, and that movement introduces the concept of time.

I mentioned that I considered the a priori assumptions about time and infinity to be relevant to the Mind of God paradox, and I chose to begin this critical examination of the relevant assumptions underlying the systems of logic we know as mathematics and science by first looking at assumptions about the existence and nature of infinity. In the next post, I plan to continue with a critical examination of our assumptions about time.

ERC – 7/7/2022


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