What Do Gödel’s Incompleteness Theorems Truly Mean? | Quanta Magazine
An editorially independent publication supported by the Simons Foundation.
Follow Quanta
Youtube
RSS
Newsletter
Get the latest news delivered to your inbox.
Subscribe
Recent newsletters
Gift Store
Shop Quanta gear
Type search term(s) and press enter
What are you looking for?
Search
Home
What Do Gödel’s Incompleteness Theorems Truly Mean?
Comment
Save Article
Read Later
Share
Copied!
Copy link
Ycombinator
Comment
Comments
Save Article<br>Read Later
Read Later
Qualia
What Do Gödel’s Incompleteness Theorems Truly Mean?
By
Natalie Wolchover
May 18, 2026
At 25, Kurt Gödel proved there can never be a mathematical “theory of everything.” Columnist Natalie Wolchover explores the implications.
Comment
Save Article
Read Later
Kurt Gödel Papers, the Shelby White and Leon Levy Achives Center, Institute for Advanced Study; Samuel Velasco and Michael Kanyongolo/Quanta Magazine
By Natalie Wolchover
Columnist
May 18, 2026
View PDF/Print Mode
continuum hypothesis
foundations of mathematics
mathematics
proofs
Qualia
set theory
All topics
In 1931, by turning logic on itself, Kurt Gödel proved a pair of theorems that transformed the landscape of knowledge and truth. These “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete. There will always be true mathematical statements that don’t logically follow from those axioms.
I spent the early weeks of the Covid pandemic learning how the 25-year-old Austrian logician and mathematician did such a thing, and then writing a rundown of his proof in fewer than 2,000 words. (My wife, when I reminded her of this period: “Oh yeah, that time you almost went crazy?” A slight exaggeration.)
p]:my-6 [&>ul]:my-6 [&>ol]:my-6 [&>p]:text-3-5 [&>p]:leading-6.5 [&>li]:text-3-5 [&>li]:leading-6.5 [&_img.alignleft]:float-left [&_img.alignleft]:mr-5 [&_img.alignleft]:ml-0 [&_img.alignleft]:my-5 [&_img.alignright]:float-right [&_img.alignright]:ml-5 [&_img.alignright]:mr-0 [&_img.alignright]:my-5 [&_figure]:m-0 [&_figcaption]:relative [&_figcaption]:flex [&_figcaption]:flex-col [&_figcaption]:gap-2 [&_figcaption]:pt-2 [&_figcaption]:pb-4-5 [&_figcaption]:mt-0 [&_figcaption]:mb-6 &_figcaption]:font-pangram [&_figcaption]:after:content-[""] [&_figcaption]:after:absolute [&_figcaption]:after:bottom-0 [&_figcaption]:after:w-11 [&_figcaption]:after:h-0.5 [&_figcaption]:after:bg-gray-1a1 [&_.caption]:block [&_.caption]:font-pangram [&_.caption]:text-0xxs [&_.caption]:leading-4-5 [&_.caption]:m-0 [&_.attribution]:block [&_.attribution]:font-pangram [&_.attribution]:text-xs [&_.attribution]:leading-4-5 [&_.attribution]:m-0 [&_.attribution]:before:content-none show-dropcap" style="color: #000000;"><br>I n philosophy, “qualia” refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted. Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it. In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.
But even after grasping the steps of Gödel’s proof, I was unsure what to make of his theorems, which are commonly understood as ruling out the possibility of a mathematical “theory of everything.” It’s not just me. In Gödel’s Proof (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R. Newman wrote that the meaning of Gödel’s theorems “has not been fully fathomed.”
Maybe not, but six decades have passed since then. Where are we with these ideas today? Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness. They had plenty to say about the implications of Gödel’s strange intellectual achievement and how it changed the course of humanity’s unending search for truth.
PANU RAATIKAINEN , philosopher at Tampere University and author of the Stanford Encyclopedia of Philosophy entry on Gödel’s incompleteness theorems
Ever since the ancient Greeks, the axiomatic method has been widely taken as the ideal way of organizing scientific knowledge. The aim is to have a small number of “self-evident” basic propositions — axioms, principles, or laws — such that all truths of the field in question can be logically derived from them.
Gödel’s incompleteness theorems show with mathematical precision that this ideal necessarily fails for large parts of mathematics. The whole of mathematical truth concerning even just positive integers (1, 2, 3 …) is so perplexingly complex that it does not follow from any finite set of axioms.
This means that some mathematical problems are not even in principle solvable by our current mathematical methods. Progress may...