Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century

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

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.

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.


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.

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.

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

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250 Responses

  1. Conrad says:

    I think Gödel’s theorem (after Penrose) might be more damaging to, say, a computational theory of consciousness than to atheism. Any algorithms used to show that consciousness is a type of strong-AI will entail other propositions that the algorithm itself cannot be used to demonstrate.

  2. John Rarere says:

    Actually.. When we go outside time and space we cannot go outside time and space anymore

  3. Filbert Justin says:

    Can Gödel’s theorem of incompleteness be applied to evolutionary theory to show that it is incomplete and then disprove it? I also knew that Godel did not believe in evolutionary theory. Here is his quotes:
    1/ From Rebbeca Goldstein’s Incompleteness: The Proof and Paradox of Kurt Gödel:

    “John Bahcall was a promising young astrophysicist when he was introduced to Gödel at a small Institute dinner. He identified himself as a physicist, to which Gödel’s curt response was `I don’t believe in natural science.’

    The philosopher Thomas Nagel recalled also being seated next to Gödel at a small gathering for dinner at the Institute and discussing the mind-body problem with him, a philosophical chestnut that both men had tried to crack. Nagel pointed out to Gödel that Gödel’s extreme dualist view (according to which souls and bodies have quite separate existences, linking up with one another at birth to conjoin in a sort of partnership that is severed upon death) seems hard to reconcile with the theory of evolution. Gödel professed himself a nonbeliever in evolution and topped this off by pointing out, as if this were additional corroboration for his own rejection of Darwinism: `You know Stalin didn’t believe in evolution either, and he was a very intelligent man.’

    `After that,’ Nagel told me with a small laugh, `I just gave up.’

    The linguist Noam Chomsky, too, reported being stopped dead in his linguistic tracks by the logician. Chomsky asked him what he was currently working on, and received an answer that probably nobody since the seventeenth-century’s Leibniz had given: `I am trying to prove that the laws of nature are a priori.’ ”

    2/ Gödel quotation from “A Logical Journey”

    “I don’t think the brain came in the Darwinian manner. In fact, it is disprovable. Simple mechanism can’t yield the brain. I think the basic elements of the universe are simple. Life force is a primitive element of the universe and it obeys certain laws of action. These laws are not simple, and they are not mechanical.”

    3/ Section 6.2.11 from A Logical Journey by Hao Wang, MIT Press, 1996:
    “I believe that mechanism in biology is a prejudice of our time which will be disproved. In this case, one disproof, in my opinion, will consist in a mathematical theorem to the effect that the formation within geological times of a human body by the laws of physics (or any other laws of a similar nature), starting from a random distribution of the elementary particles and the field, is as unlikely as the separation by chance of the atmosphere into its components.”

    4/ Section 6.2.13 from A Logical Journey by Hao Wang, MIT Press, 1996:
    “Darwinism does not envisage holistic laws but proceeds in terms of simple machines with few particles. The complexity of living bodies has to be present either in the material or in the laws. The materials which form the organs, if they are governed by mechanical laws, have to be of the same order of complexity as the living body.”

  4. Ralph Shumaker says:

    I had a hard time finding this article again amidst the plethora of matches from an internet search. I’m glad it’s still here. I want to copy this article for my own use (and the address with it).

    I had forgotten a few of the points you made in this excellent article. I suspect there are far more applications of it than just the few you mentioned. For example, monotheism often degenerates into polytheism, but never the other way around. And society always moves farther away from God, never closer. I suspect that it could even prove that good is outside the circle of evil.

    These are just a few of the ideas that I wish to pursue with it.

  5. Edward Price says:

    Hi, Perry.

    There are lots of comments here, so it’s possible that I will end up repeating what someone else has already said. Please forgive me if this is the case.

    I think that your attempt to apply Gödel’s Incompleteness Theorems here is faulty.

    First, I take issue with the way(s) you describe Gödel’s Incompleteness Theorems (there are two of them!). One thing that it seems you state here is that Gödel’s Incompleteness Theorems show that one cannot prove something without an axiom. “Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.”

    This is not what Gödel’s Incompleteness Theorems state. Rather, this is an immediate consequence of the definition of “proof” used in formal logic. A proof is a finite sequence of sentences in which every sentence is either an axiom or follows from previous sentences by an established rule of inference. It is immediately clear from this definition that without axioms, nothing can be proven. There’s also a bit of a weird technicality here about axioms and provability. In this formal logical setting, axioms are technically considered “provable”. They can be proved by the one-line proof which consists of simply stating the axiom itself.

    Looking at Gödel’s own paper where he published his incompleteness theorems, he gives the following definition of proof:
    “Die Klasee der beweisbaren Formeln wird definiert als die kleinste Klasse von Formeln, welche die Axiome enthält und gegen die Relation “unmittelbare Folge” abgeschlossen ist.”
    This translates to English (via Google translate) as:
    “The class of provable formulas is defined as the smallest class of formulas which contains the axioms and is closed to the relation “immediate consequence”.”

    Again, from Gödel’s own definition of proof (from the setup of terminology and notation in his paper – before he even gets to the content of his paper), we see these two points. Gödel considers axioms themselves to be provable. Moreover, without any axioms, the set of provable formulas becomes the empty set, as the empty set would then contain all axioms and is vacuously closed under “immediate consequence”.

    So no, Gödel’s theorems do not prove that one needs to have some unproved assumptions. If that was the content of Gödel’s proofs, no one would ever know about him today.

    (This discrepancy between your notion of axioms and Gödel’s notion of axioms should give you pause! Consider the possibility that you don’t actually understand Gödel’s work as well as you think you do, if you’re disagreeing with Gödel on the basic terminology involved!)  

    Next, the circle thing. Yes, you are trying to make the statement accessible to non-experts, but the circle analogy really stretches Gödel’s theorems far beyond what they actually say. Gödel’s theorems do not prove the existence of objects outside of any classification of objects one could come up with. Rather, Gödel’s First Incompleteness Theorem states that if you have a list of axioms where you are able to determine what each axiom is and if this list of axioms is powerful enough to describe the arithmetic of the natural numbers, then that set of axioms is either inconsistent (meaning it can prove contradictory statements) or is incomplete (meaning that there is a sentence where neither the sentence itself nor the sentence’s negation is provable from those axioms).

    A more apt circle analogy for Gödel’s First Incompleteness Theorem is that if you put “in a circle” every sentence that is provable or refutable (meaning its negation is provable) from a set of known axioms, then either that set of axioms is inconsistent or there exists a sentence outside of that circle.

    By using the “circle” analogy to talk about objects such as bicycles, you’re really stepping into awkward territory here. In what way does a bicycle represent the collection of all sentences provable or refutable from an axiom set capable of describing natural number arithmetic? It’s not at all clear that Gödel’s theorem should apply here. Note: I’m not saying that there doesn’t exist something other than the bicycle. I’m just saying that it’s not accurate to say that the existence of something other than the bicycle is *proven by Gödel’s incompleteness theorems*. I see no clear reason why Gödel’s theorems should apply to physical objects. And if there’s no reason why Gödel’s theorems should apply to this sort of thing, then your argument fails before it even begins.

    But let’s say that for sufficiently large amounts of objects (“universe” scale), you are able to make some weird connection between this large collection of objects and the sentences provable from some axiom set. In order to apply Gödel’s theorems to this situation, you would still have the work of showing that the axiom set satisfies the hypotheses of Gödel’s theorems. I’ve seen you in a video before where you declare that Gödel’s theorems apply to everything! But this is demonstrably false. Gödel’s theorems apply to axiomatic systems which satisfy the hypotheses of his theorems.

    In particular, you would need to show that this axiom set associated with the objects of the universe is effectively generated (meaning we are capable of identifying all axioms), is able to describe enough of natural number arithmetic, and is consistent (if you wanted to show it is incomplete). *All* of these hypotheses are important. For each hypothesis, we have examples of axiom sets that satisfy all the hypotheses except that one and in which the conclusions of Gödel’s incompleteness theorems are false.

    For example, one can get a complete, consistent theory of the natural numbers which is capable of fully describing the arithmetic of the natural numbers. To do this, simply take all true statements about the natural numbers as axioms. The problem is that this axiom set is not effectively generated. This is something you have not addressed – that “the universe” is effectively generated. And this actually shows how far you’ve stretched Gödel’s theorems past what they actually say. What in the world does it mean to say that the circle around “the universe” is effectively generated? The question is nonsensical. But in order to apply Gödel’s theorems in the way you’re trying to apply them, you have to be able to answer this question.

    Next, we move on to being powerful enough to describe the arithmetic of natural numbers. You claim that this is possible in the universe, but I’m not so sure. In the observable universe, there are only finitely many “things” (objects, both concrete and abstract), space, and time, as you yourself admit. But in order for Gödel’s proof to work, you need an infinite amount of something. If you actually understand Gödel’s proof (not the theorem statement, but the proof itself), then this is clear. Gödel’s proof critically relies on being able to embed the sentences and proofs of a theory into the natural numbers in a recursive manner. The collection of sentences and proofs is necessarily infinite, however. So in order to successfully do a Gödel embedding, you need infinitely many “things”. Without a Gödel embedding, Gödel’s proof technique completely fails since it relies fundamentally on the existence of such an embedding.

    I have seen that you like to argue that the universe is capable of describing the natural number arithmetic because Euclidean geometry can and because the universe is more complex than Euclidean geometry. This is a misunderstanding. If you establish the postulates that Euclid used, then yes, Euclidean geometry is capable of describing natural number arithmetic. It does this through the use of line segments, lines, and circles. Each natural number can be described because Euclidean geometry is capable of producing arbitrarily long line segments. If we were to give a global cap on the length of line segments, then not all natural numbers could be described. So even though the universe might possibly be more “complex” than Euclidean geometry in some sense, this does not imply that the observable universe is capable of describing natural number arithmetic.

    I won’t get into consistency of the universe. People could argue that the universe is inconsistent, and that may ultimately be true. I don’t know. But that’s about all I can say on this point.

    So, in order to apply Gödel’s theorems to something like “the universe”, you would have to show all of the following non-obvious assertions:
    1. Each “thing” in the universe can be corresponded to a statement in a formal language that can be proven or refuted by some collection of axioms so that
    2. We are able to identify whether a statement in the formal language is or is not an axiom, 
    3. This axiom system is capable of describing natural number arithmetic, and
    4. This axiom system is consistent.

    Assertion 1 itself seems hopeless to me, let alone the other three.

    • “1. Each “thing” in the universe can be corresponded to a statement in a formal language that can be proven or refuted by some collection of axioms so that…..”

      Is not the above statement precisely what science is attempting to do and be — a consistent system of laws, expressed in terms of numbers and arithmetical functions, that models the behavior of objects in the universe?

      If the rules of algebra (or addition or commutative or associative properties etc) apply to numbers – and therefore also apply to calculations about falling objects – then please explain to me why Godel’s theorems do not similarly apply to the same.

      You are welcome to reject the notion that mathematical truths apply to the universe, but in doing so you are rejecting the very premise of science itself.

      If you choose to do that, that is your decision.

  6. Lukasz Stepien says:

    Referring to the issue of the provability of consistency of the Arithmetic System, I’d like to inform you about the paper: T. J. Stępień, Ł. T. Stępień, „On the Consistency of the Arithmetic System”, Journal of Mathematics and System Science, vol. 7, 43 (2017), arXiv:1803.11072 , where a proof of the consistency of the Arithmetic System has been published. This proof has been done within this Arithmetic System.

  7. Toni Segarra says:

    We cannot know everything. Because, everything is infinite. And the mind is too. So, whatever is said, both can be denied and affirmed infinitely.

  8. Marcel Kincaid says:

    The stunningly absurd argument that atheism violates the laws mathematics was posted at Quora as an example of the silliest misapplications of Gödel’s Incompleteness Theorem.

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