Gödel’s Incompleteness Theorem:

The #1 Mathematical Discovery of the 20th Century

In 1931, Kurt Gödel delivered a devastating blow to the mathematicians of his time

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

Gödel’s discovery not only applied to mathematics but literally all branches of science, logic and human knowledge. It has truly 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 list 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 reasonable that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that they 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 many things that were “obviously” true but nobody could figure out a way to prove them.

In the early 1900’s, however, a tremendous sense of optimism began to grow in mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) were 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.

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 discovered 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.”

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

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.

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. Incompleteness is true in math; it’s equally true in science or language or philosophy.

And: If the universe is mathematical and logical, Incompleteness also applies to the universe.

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.

So Gödel, in one of the most ingenious moves in the history of math, converted the Liar’s Paradox into a mathematical formula. He proved that any statement requires an external observer.

No statement alone can completely prove itself true.

His Incompleteness Theorem was a devastating blow to the “positivism” of the time. 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 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 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 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 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. You cannot prove that the universe is rational. You can only observe that mathematical formulas like E=MC^2 do seem to perfectly describe what the universe does.

All predictions about the future are inductive. Outside the circle. In Gödel’s language they are “undecidable propositions.” It’s probable you’ll still have your job next week… but maybe you don’t.

Nearly all scientific laws are based on inductive reasoning. These laws rest on an assumption that the universe is logical and 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.)

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 13.7 billion years time
• The universe is mathematical. Any physical system subjected to measurement performs arithmetic. (You don’t need to know math to do addition – you can use an abacus instead and it will give you the right answer every time.)
• The universe (all matter, energy, space and time) cannot explain itself
• Whatever is outside the biggest circle is boundless. 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. 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 3.5 billion years ago (Or was it longer? Was information somehow present at the beginning?). It came in the form of the Genetic code, which is symbolic and immaterial.
• The information appears to have 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.

My book Evolution 2.0: Breaking the Deadlock Between Darwin and Design explores the Origin of Information question in depth. The Evolution 2.0 Prize offers a multi-million dollar award for Origin of Information.

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 conscious.

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

So 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 that all logical systems must rely on something outside the system. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct. If the universe is logical, it has an outside cause.

Atheists do have an “out” on this. Atheists still have the option to insist that there is nothing outside the universe. But if that’s true, then the universe itself is illogical. Therefore science itself is invalid.

So you can have atheism or you can have science – but you can’t have both.

This is why atheism violates the laws of reason and logic.

Gödel’s Incompleteness Theorem definitively proves that science can never fill its own gaps. We have no choice but to look outside of science for answers.

The Incompleteness of the universe isn’t proof that God exists. But… it IS proof that in order to construct a rational, scientific 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. It’s 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 science and mathematics.

Perry Marshall

“Without mathematics we cannot penetrate deeply into philosophy.
Without philosophy we cannot penetrate deeply into mathematics.
Without both we cannot penetrate deeply into anything.”

-Leibniz

I snapped a photo of this quote at the Galileo Museum in Florence Italy.

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 on Wikipedia

Science vs. Faith on CoffeehouseTheology.com

Information Theory: “If you can read this, I can prove God exists”