In 1931, the young mathematician Kurt Gödel made a landmark discovery, as powerful as anything Albert Einstein developed.

In one salvo, he completely demolished an entire class of scientific theories.

Gödel’s discovery not only applies to mathematics but literally all branches of science, logic and human knowledge. It has earth-shattering implications.

Oddly, few people know anything about it.

Allow me to tell you the story.

Mathematicians love proofs. They were hot and bothered for centuries, because they were unable to PROVE some of the things they knew were true.

So for example if you studied high school Geometry, you’ve done the exercises where you prove all kinds of things about triangles based on a set of theorems.

That high school geometry book is built on Euclid’s five postulates. Everyone knows the postulates are true, but in 2500 years nobody’s figured out a way to prove them.

Yes, it does seem perfectly “obvious” that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that Euclid’s postulates are a reasonable, and in fact *necessary*, set of 5 assumptions.

Towering mathematical geniuses were frustrated for 2000+ years because they couldn’t prove all their theorems. There were so many things that were “obviously true,” but nobody could find a way to prove them.

In the early 1900’s, however, a tremendous wave of optimism swept through mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) became convinced that they were rapidly closing in on a final synthesis.

A unifying “Theory of Everything” that would finally nail down all the loose ends. Mathematics would be complete, bulletproof, airtight, triumphant.

In 1931 this young Austrian mathematician, Kurt Gödel, published a paper that once and for all PROVED that a single Theory Of Everything is actually impossible. He *proved* they would never *prove* everything. (Yeah I know, it sounds a little odd, doesn’t it?)

Gödel’s discovery was called “The Incompleteness Theorem.”

If you’ll give me just a few minutes, I’ll explain what it says, how Gödel proved it, and what it means – in plain, simple English that anyone can understand.

**Gödel’s Incompleteness Theorem says:
**

**“Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.”**

You can draw a circle around all of the concepts in your high school geometry book. But they’re all built on Euclid’s 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.

Stated in Formal Language:Gödel’s theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.” The Church-Turing thesis says that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness. Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.) Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete. Syllogism: 1. All non-trivial computational systems are incomplete 2. The universe is a non-trivial computational system 3. Therefore the universe is incomplete |

You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.

You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.

**Gödel proved that there are ALWAYS more things that are true than you can prove. **Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.

Gödel’s Incompleteness Theorem applies not just to math, but to *everything* that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it’s equally true in science or language and philosophy.

Gödel created his proof by starting with “The Liar’s Paradox” — which is the statement

**“I am lying.”
**

“I am lying” is self-contradictory, since if it’s true, I’m not a liar, and it’s false; and if it’s false, I am a liar, so it’s true.

Gödel, in one of the most ingenious moves in the history of math, converted this Liar’s Paradox into a mathematical formula. He proved that no statement can prove its own truth.

You always need an outside reference point.

The Incompleteness Theorem was a devastating blow to the “positivists” of the time. They insisted that literally anything you could not measure or prove was nonsense. He showed that their positivism was nonsense.

Gödel proved his theorem in black and white and nobody could argue with his logic. Yet some of his fellow mathematicians went to their graves in denial, believing that somehow or another Gödel must surely be wrong.

He wasn’t wrong. It was really true. There are more things that are true than you can prove.

A “theory of everything” – whether in math, or physics, or philosophy – will never be found. Because it is mathematically impossible.

OK, so what does this really mean? Why is this super-important, and not just an interesting geek factoid?

Here’s what it means:

**Faith and Reason are not enemies.**In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you*cannot prove.***All closed systems depend on something outside the system.****You can always draw a bigger circle but there will still be something outside the circle.**

**Reasoning inward from a larger circle to a smaller circle (from “all things” to “some things”) is deductive reasoning.**

Example of a deductive reasoning:

1. All men are mortal

2. Socrates is a man

3. Therefore Socrates is mortal

**Reasoning outward from a smaller circle to a larger circle (from “some things” to “all things”) is inductive reasoning.**

Examples of inductive reasoning:

1. All the men I know are mortal

2. Therefore all men are mortal

1. When I let go of objects, they fall

2. Therefore there is a law of gravity that governs all falling objects

Notice than when you move from the smaller circle to the larger circle, you have to make assumptions that you cannot 100% prove.

For example you cannot PROVE gravity will always be consistent at all times. You can only observe that it’s consistently true every time.

Nearly all scientific laws are based on inductive reasoning. **All of science rests on an assumption that the universe is orderly, logical and mathematical based on fixed discoverable laws.**

You cannot PROVE this. (You can’t prove that the sun will come up tomorrow morning either.) You literally have to take it on faith. In fact most people don’t know that outside the science circle is a philosophy circle. Science is based on philosophical assumptions that you cannot scientifically prove. Actually, the scientific method cannot prove, it can only infer.

(Science originally came from the idea that God made an orderly universe which obeys fixed, discoverable laws – and because of those laws, He would not have to constantly tinker with it in order for it to operate.)

**
Now please consider what happens when we draw the biggest circle possibly can – around the whole universe.** (If there are multiple universes, we’re drawing a circle around all of them too):

**There has to be something outside that circle.**Something which we have to assume but cannot prove**The universe as we know it is finite**– finite matter, finite energy, finite space and 13.8 billion years time**The universe**(all matter, energy, space and time)**cannot explain itself****Whatever is outside the biggest circle is boundless.**So by definition it is not possible to draw a circle around it.- If we draw a circle around all matter, energy, space and time and apply Gödel’s theorem, then we know what is outside that circle is not matter, is not energy, is not space and is not time. Because all the matter and energy are inside the circle.
**It’s immaterial.** **Whatever is outside the biggest circle is not a system**– i.e. is not an assemblage of parts. Otherwise we could draw a circle around them. The thing outside the biggest circle is*indivisible*.**Whatever is outside the biggest circle is an uncaused cause,**because you can always draw a circle around an effect.

**We can apply the same inductive reasoning to the origin of information:**

**In the history of the universe we also see the introduction of information,**some 3.8 billion years ago. It came in the form of the Genetic code, which is symbolic and immaterial.**The information had to come from the outside,**since information is not known to be an inherent property of matter, energy, space or time.**All codes we know the origin of are designed by conscious beings.****Therefore whatever is outside the largest circle is a conscious being.**

When we add information to the equation, we conclude that not only is the thing outside the biggest circle infinite and immaterial, it is also self-aware.

Isn’t it interesting how all these conclusions sound suspiciously similar to how theologians have described God for thousands of years?

Maybe that’s why it’s hardly surprising that 80-90% of the people in the world believe in some concept of God. Yes, it’s intuitive to most folks. But Gödel’s theorem indicates it’s also supremely *logical*. In fact it’s the only position one can take and stay in the realm of reason and logic.

The person who proudly proclaims, “You’re a man of faith, but I’m a man of science” doesn’t understand the roots of science or the nature of knowledge!

Interesting aside…

If you visit the world’s largest atheist website, Infidels, on the home page you will find the following statement:

**“Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it.”
**

If you know Gödel’s theorem, you know all systems rely on something outside the system. So according to Gödel’s Incompleteness theorem, the folks at Infidels cannot be correct. Because the universe is a system, it has to have an outside cause.

Therefore Atheism violates the laws mathematics.

The Incompleteness of the universe isn’t proof that God exists. But… it IS proof that in order to construct a consistent model of the universe, belief in God is not just 100% logical… it’s necessary.

Euclid’s 5 postulates aren’t formally provable and God is not formally provable either. But… just as you cannot build a coherent system of geometry without Euclid’s 5 postulates, neither can you build a coherent description of the universe without a First Cause and a Source of order.

**Thus faith and science are not enemies, but allies.** They are two sides of the same coin. It had been true for hundreds of years, but in 1931 this skinny young Austrian mathematician named Kurt Gödel proved it.

No time in the history of mankind has faith in God been more reasonable, more logical, or more thoroughly supported by rational thought, science and mathematics.

Perry Marshall

** “Math is the language God wrote the universe in.” –**Galileo Galile, 1623

Further reading:

*“*Incompleteness: The Proof and Paradox of Kurt Gödel” by Rebecca Goldstein – fantastic biography and a great read

A collection of quotes and notes about Gödel’s proof from Miskatonic University Press

Formal description of Gödel’s Incompleteness Theorem and links to his original papers on Wikipedia

Science vs. Faith on CoffeehouseTheology.com

Hi Perry,

I have read through a lot of the posts and I strongly agree that there are gaps in the reasoning regarding Gödel. Gödel’s incompleteness theorems relate to pure mathematics – Formal language, Natural numbers which includes a arithmetic axioms which as far as I can understand have no real impact on physics (the strict axioms are related to what is called Paeno Arithmetic). The main problem with Paeno Arithmetic is that it includes the true infinite. Gödel’s approach was to try to prove PA with finite means. What he found was his incompleteness theorem which basically misses that. While it certainly totally destroyed David Hilbert, Bertrand Russell’s and others who hoped to create a perfect formal mathematical system, I have hard to understand why it would have any impact on more trivial arithmetic systems which we use for physics.

In addition Gödel’s proof for the first theorem (the second is derived from the first) is based on contradiction (not deduction as you said in one of your posts), and yes he uses a paradoxical statement such as; “This statement is not a theorem in the System”. Which if it is false is still a theorem inside the system -> In which case the statement is valid but false. This means the statement must be true. If it is true then it simply can’t be verified within the system. So what Gödel says is that in a formal language system with the strictness of the Paeno Arithmetic axioms then there must be possible to create statements that are true but which cannot be proven inside this system. As far as I can see there is no limit on systems in which you cannot make such statements. This type of statements as the one Gödel used to prove his theorem would be utterly nonsense for any other purpose than to prove the theorem. One more note on formal systems.

And just because this sounds like a generic logical statement the meaning of a formal language and Paeno arithmetic is essential for the proof and I have very hard to see how this theorem can be used to jump to far-reaching conclusions about the universe, especially in case the universe is finite. There are examples of arithmetic problems (such as Fermat’s Conjecture) which had been suggested to possibly be of the type suggested by Gödel as true but unsolvable inside the system, which have later been solved using other mathematical methods than Arithmetic based on Paeno Arithmetic (but the fact is we don’t know if it would be possible to prove it within the system).

In his book “Gödel’s Theorem” Torkel Franzen makes a point that “Nothing in the incompleteness theorem excludes the possibility of our producing a compete theory of stars, ghosts and cats, all rolled into one, as long as what we say about stars, ghosts and cats can’t be interpreted as a statement about the natural numbers”. In fact most natural science does not depend on the higher arithmetic’s included in PA, but only on simple arithmetic, calculus, algebra and geometry which Gödel’s theorems has nothing to say about.

For more details about Gödel’s theorem and one source for some of my writings above I suggest reading //math.stanford.edu/~feferman/papers/Godel-IAS.pdf

Torkel Franzen, “Gödel’s Theorem” which at length discusses what Gödel’s theorem cannot be used for.

To conclude

I have a problem that you only use Gödel’s general description leaving out what the theorem or its proof really says. In particular I have a problem with the jump from mathematical or logical theory to applying the laws to the Universe itself (Apart from that I don’t understand the jump, this is epistemologically difficult, since it implies a lot of assumptions about our ability to know the true nature of the Universe which are not given that everyone would agree with you about). Given how little we know about anything we have to be a bit more humble than that. The Incompleteness Theorems, still being very important, probably have very little to say about the existence of God

But even if we for the sake of the argument would assume that Gödel applies for consistent formalised belief systems. In which case the atheist and the theist probably are on equal ground, since the statement if God exists or not cannot still be proved inside the system, hence it would still be a matter of faith. i.e. “God Exist which cannot be proven inside the system” or “God does not exist which cannot be proven inside the system” are in my understanding equal and if you assume one of them to be true you must just agree that it cannot be proven inside the system. This of course would (still assuming that Gödel applies) mean that the Atheists would have to agree that science would never be able to prove God’s non-existence.

Karkywo, one thing is certain and that is Perry does not understand Gödel. You have a better take on it. The following is the central part of the theorem.

Axioms are the building premises that all mathematical systems are based on. These are selected for their intuitive correctness. Gödel theorem only goes to the incompleteness of systems that are defined as “formal systems.” Incompleteness means there will always be an axiom within such as system that cannot be proven by its own operations.

Here are the three features/conditions of formal systems; and it is only when all are present that a system is incomplete according to Godel’s theorem.

(1) The system must be finitely specified i.e. there is not an incomputable infinity of axioms.

(2) The system must be large enough to include all the symbols and axioms used in Peano arithmetic. It is only when a system includes addition and multiplication that incompleteness emerges; therefore algebra is incomplete. Simpler mathematical systems; such as, Presburger Arithmetic (does not include multiplication) and Euclidean, non-Euclidean Geometry are not formal systems and are complete.

(3) The system must be consistent i.e. it cannot contain contradictory axioms or operations.

Godel’s theorem has nothing to say about physics. Gödel, for example, did not associate his theorem with Heisenberg’s Uncertainty Principle or anything in quantum mechanics. It also has absolutely nothing to do with God. (See: Barrow, John D., New Theories of Everything, Oxford University Press, Oxford, New York, 2007, pp. 51-61.)

You are right and wrong that Godel Incompleteness has nothing to do with physics. Certainly it has nothing to do with the current state of physics. But physics is a science that changes by paradigm shifts, as it did with Newton, then with Relativity and Quantum Mechanics.

I think the next paradigm shift in Physics will very much involve Godel Incompleteness as a mathematical mechanism by which order arises out of disorder in self organizing systems.

Basically I’m saying that the next revolution in physics will involve explaining the Big Bang as a self organizing system.

FYI: you said: Incompleteness means there will always be an axiom within such as system that cannot be proven by its own operations.

I think you mean there will be a THEOREM that cannot be proven.

I have a direct comment regarding this article (my previous comment was not directly about the OP’s article, though it was about Godel Incompleteness).

The OP equates god to that “thing” which is always outside the circle. The circle he is referring to has nested circles within it. Each circle is that formal system in which you can prove the itheorems of the nested circles within it.

I propose that he has the tail wagging the dog. That thing which is outside is not god (or the proto formal system in more mathematical terms). The outside circles are emanations, not original creators. If you want to invoke the idea of god in Godel Incompleteness then god is the innermost circle (first formal system) not the outermost circle, which is just an emanation from god.

What an excellent post, Karkywo! I agree that there is no way to prove that God exists or doesn’t exist. Any “proof” would have to be reduced to terms and concepts that our humble minds can handle. Reduction means that we are not talking about God any more – we our talking about ourselves.

Have you read anything on the Godel – Wittgenstein conflicts?

Some people have said that God is mathematics. All they are really saying is, “When I think about God, I think math.”

Atheists are too often engaged in the fool’s game of trying to excluse God. I think that we, as believers, are too often tempted to try to squeeze God into the narrow confines of our concept-forming abilities, be they mathematical or philosophical. I don’t include science, because science has nothing to say about God, however much we would like it to. Even God doesn’t have much to “say” about God if you compare the Holy Scriptures to the infinite (in-conceivable) nature of God.

[…] I was digging through articles for this blog, I found this gem illustrating a stunning abuse of the incompleteness theorem in an attempt to prove the existence of […]

The OP doesn’t understand anything to what the Gödel’s Incompleteness Theorem is. He is ridiculous on a mathematical point of view, from the mention of the Euclidian postulates to the end of the post…

Source: ask your preferred mathematician !

This is mathematics for the dummies… by a dummy.

I suddenly encountered somewhere in the article about the inconsistency of gravity. True, even strengthened by the recent discovery of Gravitational waves.

Is Gödel’s logic God’s logic?

You can ask that of anyone’s logic. Can you detect flaws in it?

While I would say that the proof of Fermat’s last theorem is #1; I will not quibble over this. Godel’s proof is, indeed, a very significant advance, and well worth the time and effort in studying it.

I would like to also read posts regarding the book “Godel, Esher, Bock” and its author’s comments on Godel’s theorem.

I have known this for years but never thought it important enough.

This knowledge doesn’t feed us, or keep us warm.

It’s just a fact which has no practical use!

I haven’t read all the comments so i don’t know if someone already said this, but the entire theory relies upon the concept of infinity, yet infinity cannot be proven, nor disprove, because it is an abstract concept, which brings me to my second objection, it is fallacious to use an abstract concept as if it were a real concept. all that being said I do realize that infinity serves a mathematical purpose, much like imaginary numbers do, and we are forced to use these abstract concepts as if they were absolutes. So in short, godel proved nothing at all, his own theory disproves its’ self.

All of logic is abstract concepts. All of mathematics is abstract concepts. I would not use this argument if I were you.

Thank you, Mr. Marshall, for your work, and for your attention to the comments in this post. (I’ve learned a lot from both you and your commenters!) Thank you for the 3 free chapters or your book; I look forward to getting into them.

After 16 years in campus ministry (my background is also engineering, btw) I’m finally going back to school (online) to learn how to think. I found this post fascinating.

Gödel’s Incompleteness Theorem is similar to Anselm’s Ontological argument in that it is incredibly simple (I marveled, for example, at how it could take Ravi Z a whole book, 239 pages, to answer “Who Made God?” until I realized that he actually answers 100 other questions.) Yet it is also nevertheless hotly debated.

How can some people find it so clearly convincing as to be practically self-evident, while others dispute it as meaningless if for no other reason (I suppose) than for its simplicity? It makes me question my own ability to recognize sound logic.

My question is: How much reading, thinking and debating must a person (you) do to be so certain of the soundness of an argument like this?

May God powerfully use this book. How is your brother doing lately?

Dan,

In my book, the way you find out if an argument is sound is: You put it out there on an anvil and let other people pound on it. Between this page and an almost identical article at //www.perrymarshall.com/godel I’ve had something like 1000 comments and nobody’s punched a hole in it. You can follow the threads blow by blow and I think you’ll find it clarifies the issues.

Bryan is still exploring and we discuss this stuff from time to time. It’s always friendly. A few years ago he said “Thank you for not letting me become an atheist.”

The only problem with your theroy is if everything has to have something above it then so does God and god’s God and god’s god’s God and so on and so forth where does it end and does that really make God a god at all

This objection is addressed repeatedly throughout this thread.

So, the only true straight line is a circle and the only true circle is a straight line?

Godel is an interesting name and word.

God = El, El = God.

El is short for Elohim, Elohim is the name of God and also the name of the Gods; The Father, The Son and The Holy Ghost.

My dear brother Perry,

I’m amazed you didn’t stop responding after some time. I know it can be exhausting. You have done an outstanding job in answering.

Thanks! truly enlightening

Hello, my comment to all…”What am I thinking right now?” Do you know anyone that can read your mind? Why is this? How come I can create a thought and you don’t know what it is, unless I tell you? And, if I tell you what I am thinking, can I change my answer to have you hear some other thought I make up? Is this act for my own amusement to show how powerless I can make you appear in my mind? Is this Logical? Where is God in your thoughts? Do we need a God at all? Do we need to create our thoughts and how be it that we can create? Why do we try to question our own existence? Who are we to be wise in our own selves? Where does such wisdom come from? Who created Wisdom? Who gave Wisdom a definition? Who or What defines us as Humans? We are so creative, that logic defines us, that we ourselves become gods. Who created God? Why do we need God? Where is God? Just look at ourselves and you will see God everywhere, because we created God.

To question or discuss the creation of ourselves, makes us Gods. We individually define our own existence. Why than discuss a Supreme Creator above and beyond ourselves? Do we want more to our lives than given? Do we seek something better than this life? Do we conclude this is it? Do we wish upon a star, or just grow old and die? This is no argument, but makes a very interesting discovery or ourselves. Those who believe that there is no Supreme Creator, have nothing to lose, but to die asking questions. Those who believe in a Divine Supreme Creator, given the name of God, or whatever name you use, seeks more to your lives than this. We seek a Creator greater than our own creation in ourselves. If we can create all this we have now and there is no end to it, why not believe in the power of our source. That power is God. You don’t need God in your life, but your waiting for the ticker to stop ticking, than your life is over. But, if there is a God who powers us all with the ability to question our own existence, I want that power to stay with me forever. If I can have that power of creation, which we all are given, what greater glory to have that power forever with you after your body dies. We believers do not live by a spark of chance, but by the power of desire for more than this. Neither believer is wrong, because we all create our own gods and we all will die. The common spark in all of us, if we choose to practice its power is the ability to Love. When you love another so not to make wider space between that person and their God, and no greater space between you and your God, you will see Love. Love is God, and when you experience the Love of God within yourself, you will know that there is something greater than you in your life. Is there a God? Put away logic and seek no space between you and another and where you find love, you will have the answer if God is real.

Entropy and Godel Incompleteness are tightly related. Entropy can be formalized as the tendency in a formal system of sufficient complexity for unprovable theorems to arise.

Just as in physics where a system starts with low entropy and moves towards high entropy, in mathematial logic a formal system starts with the axioms and moves from a state of low unprovability towards high unprovability.

I am the author of Shards Of Divinities. In my novel I form a new theory of Reality using ideas from Godel Incompleteness and Chaos Theory. In fact, I unify Chaos Theory and Godel Incompleteness into one cohesive theory.

Welcome.

Calculations for said circle leave it fuzzy enough to contain the infinite. In fact. the infinite is REQUIRED for the calculations to work. Or is PI suddenly a rational number? In that light the circle being calculated from infinity itself is perfect enough to contain whatever object, no matter how vast it may be, within its circumference. PI goes on forever. In ANYTHING could contain the universe/multiverse…..its a circle.

If you’re going to write on subjects like logic, you really need to learn the difference between postulates and theorems. Postulates are true a priori: you don’t prove them, because you don’t need to prove them. They are self-evidently true. Theorems do require proof, using the definitions, postulates, and rules of inference. I quit reading when it became evident that the author did not understand this. Presuming to discuss Goedel’s Theorem while not understanding this difference is like not knowing the Russian alphabet but claiming to understand Dostoyevsky.

Any idea or theory that pretends to explain the universe, believers, atheists, nihilists, etc., everything that is said. Both can be affirmed, as can be denied, infinitely.

It’s interesting to see how a scientific argument turns into a religious one. That in itself shows an undeniable relationship and affinity between the two.

I would like to add Art to the equasion, perhaps that way we can find some harmony. Science – Religion – Art. Which relate to each one of us in our activities of Thinking – Feeling – Willing and in our physical bodies in the head & nerves system – circulatory & rhythmic system – metabolic & limb system.

Us as human being have these three constantly developing principles in us all the time. If one of these priciples falls out of balance we get ill.

Intellectually we can choose not to believe in God or have a religion, but practically that is less of an option. We need to relate and connect (relegare) to ourselves, people and nature in order to be healthy on earth and experience a certain degree of wellbeing. Starting with breathing, talking, eating and drinking. But also finer nourishments like music, painting, reading, etc.

Even if we are not religious we practice religion by the need to survive, develop and thrive.

If atheists were really coherent in their beliefs they would also have to stop believing in the need for the above mentioned necessities and activities.

I also think some of the questions have been answered by Rudolf Steiner in his book The Philosophy of Freedom. Steiner also writes about the need to expand our perceptive capacities, which would enrich our knowledge and science too, to include Imagination, Inspiration and Intuition. These are attainable for the open minded seeker free of prejudice, willing to train ones finer capacities to reach in the so called supersensible realms. Which, if I understand Perry correctly, is the realm of God.

Good comment.