In 1930, Kurt Gödel (1906–1978) presented his incompleteness theorems to the Vienna Circle, and changed the world forever.

The Vienna Circle were the leading intellectuals of their time, on the forefront of mathematical research. Their principal project was to reduce all of mathematics to fundamental principles. That would in turn allow them to automatically prove everything that is mathematically true, and disprove everything that is mathematically false. If successful, the project would be the mathematical Theory of Everything. Essentially, the Vienna circle was working hard at putting themselves out of a job.

Gödel ran up and kicked over all their mathematical sandcastles.

He did that by showing that their fundamental assumption was false. There is no single set of principles that can illuminate all mathematical truth. And Gödel demonstrated this fact by proving that there are unprovable truths. Quite the paradox!

Beyond paradox

In fact, Gödel’s proof is beyond paradox. The precise statement that Gödel proves is “This statement cannot be proven.” At this point, you may ask, isn’t the proof actually a contradiction? How can Gödel prove that something cannot be proven? To understand what’s going on, we need to look at the mechanics of his proof.

The fundamental technique Gödel uses is this: He turns proofs into numbers, and numbers into proofs. While the technique was unheard of in his day, we are very familiar with it today. It is the same technique used by computers to store information. Everything in a computer is stored using 0’s and 1’s as binary numbers. So in that respect Gödel anticipated modern computers but instead of binary numbers, Gödel used prime numbers.

Because every possible proof has a one-to-one correspondence with a number, saying that “X is not provable” is identical to saying that there is no number that represents a proof for X.

The same goes for disproof. Gödel creates a statement in this proof numbering scheme that — when fed a number N — states “N is not provable.” He then takes the number that represents the statement, and feeds it to the statement. That makes the statement self-referential. We’ll represent this self-referential statement with the letter O, because the letter O is a circle, which works somewhat like self-reference.

With the self-referential statement O in hand, we can look at any number N and ask “Does N prove O?” If it does, then since O states that it cannot be proven, we have a contradiction. Therefore, no N can prove O.

On the other hand, we can ask “Does N disprove O?” If it does, then, because O says that it cannot be proven, N at the same time becomes a proof of O — which is again a contradiction.

So, either way, for every N, a contradiction results if it proves or disproves O. Therefore, there is no proof or disproof of O. Which means… drum roll please… that O is true!

No contradiction

This is how Gödel performed his remarkable feat without creating a contradiction. There is no contradiction because the concept of “proof” is being used in two different ways. For every N applied to O, the word “proof” means that there is no finite sequence of logical principles and rules that result in either O or its negation, because every finite number N can only represent a finite proof.

To create a proof that O is true we must assemble an infinite list of all possible numbers showing that every single number results in a contradiction when applied to O. Since all N are finite then no N can create this infinite proof. On the other hand, it is easy for us readers to see that every N results in a contradiction, therefore O is true. The difference hinges on the fact that our minds can grasp infinity, whereas finite numbers can only ever create finite proofs.

From this analysis of how Gödel’s proof works, we get an interesting insight into the nature of mind. There are two major responses to Gödel’s proof. There are some who claim that his proof shows that the mind is somehow beyond finite mathematical systems, and therefore non-physical. There are others who claim the proof says nothing about our minds being non-physical, because our minds could have their own incompleteness theorems.

Both right and wrong
However, the naysayers are both right and wrong. Since O can be proven with an infinite proof, then the mind that grasps O’s truth could be an infinite number. At the same time, there is no way such a mind could exist in the finite physical world.

It is clear from Gödel’s letters to his mother that his mathematical work motivated his view on the afterlife. While not a fan of some forms of organized religion (he referenced “bad churches” in one of his letters), Gödel believed that a finite life could never actualize even a fraction of its infinite potential (as evidenced by his proof). So there must be a second life where our infinite potential is actualized: “If the world is rationally organised and has meaning, then it must be the case. For what sort of a meaning would it have to bring about a being (the human being) with such a wide field of possibilities for personal development and relationships to others, only then to let him achieve not even 1/1,000th of it?”

Addendum

For those interested in diving deeper into Gödel’s proof: I read through the English translation of his paper, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” Often, there are bigger considerations surrounding major mathematical innovations, and the best way to get the full picture is to read the original source, instead of a simplified retelling (like mine!).

Be warned! There is a small, yet fundamental, transmission error in one of the crucial theorems of his proof in that English translation. Theorem 15 is missing a bar over the left side of the formula. The bar is also missing in the second half of the undecideability proof for proposition VI. This makes the argument incoherent and confusing to follow.

A free, online English translation of Gödel’s paper can also be found here. However, I haven’t read through the translation carefully myself to be able to vouch for it. Theorem 15 is written correctly, at least. The site also has an interesting set of quotes from Gödel in favor of intelligent design. Incidentally, the website’s author, James R. Meyer, thinks that Gödel’s proof has a fundamental flaw.

You may also wish to read: Why logician Kurt Gödel believed in life after death. He saw human folly as an opportunity to reform and learn, because our souls are immortal whether we like it or not. In a deeply rational, ordered universe, Gödel argues, human potential — frustrated in so many ways here — must flower afterward elsewhere.