What is Gödel's proof?

Join Our Community of Science Lovers!

Kurt Gödel's incompleteness theorem demonstrates that mathematics contains true statements that cannot be proved. His proof achieves this by constructing paradoxical mathematical statements. To see how the proof works, begin by considering the liar's paradox: "This statement is false." This statement is true if and only if it is false, and therefore it is neither true nor false.

Now let's consider "This statement is unprovable." If it is provable, then we are proving a falsehood, which is extremely unpleasant and is generally assumed to be impossible. The only alternative left is that this statement is unprovable. Therefore, it is in fact both true and unprovable. Our system of reasoning is incomplete, because some truths are unprovable.

Gödel's proof assigns to each possible mathematical statement a so-called Gödel number. These numbers provide a way to talk about properties of the statements by talking about the numerical properties of very large integers. Gödel uses his numbers to construct self-referential statements analogous to the plain English paradox "This statement is unprovable."

On supporting science journalism

If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.

Strictly speaking, his proof does not show that mathematics is incomplete. More precisely, it shows that individual formal axiomatic mathematical theories fail to prove the true numerical statement "This statement is unprovable." These theories therefore cannot be "theories of everything" for mathematics.

The key question left unanswered by Gödel: Is this an isolated phenomenon, or are there many important mathematical truths that are unprovable?

It’s Time to Stand Up for Science

If you enjoyed this article, I’d like to ask for your support. Scientific American has served as an advocate for science and industry for 180 years, and right now may be the most critical moment in that two-century history.

I’ve been a Scientific American subscriber since I was 12 years old, and it helped shape the way I look at the world. SciAm always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does that for you, too.

If you subscribe to Scientific American, you help ensure that our coverage is centered on meaningful research and discovery; that we have the resources to report on the decisions that threaten labs across the U.S.; and that we support both budding and working scientists at a time when the value of science itself too often goes unrecognized.

In return, you get essential news, captivating podcasts, brilliant infographics, can't-miss newsletters, must-watch videos, challenging games, and the science world's best writing and reporting. You can even gift someone a subscription.

There has never been a more important time for us to stand up and show why science matters. I hope you’ll support us in that mission.

Thank you,

David M. Ewalt, Editor in Chief, Scientific American