Goedel Proved Hilbert Wrong

I wondered whether Goedel literally proved Hilbert wrong, as is commonly claimed. I doubted it.

David Hilbert didn’t leave a single, tidy quote where he explicitly says, “I demand completeness and consistency provable from within,” in those exact words. His vision for the formalization of mathematics, known as Hilbert’s Program, emerged through his lectures, papers, and talks—particularly in the 1920s — and is pieced together from these sources. I’ll show you where these demands are clearest, based on his own words and how contemporaries understood him, without inventing anything he didn’t say.

Completeness

Hilbert’s call for a complete axiomatic system — where all true statements in a mathematical domain (like arithmetic) can be derived from the axioms—shines through in his 1900 address to the International Congress of Mathematicians in Paris. There, he posed his famous 23 problems, and the second problem implicitly reflects this goal. He asked for a proof of the consistency of arithmetic’s axioms but tied it to a broader vision of a “definite system” that settles all questions. He wrote:

“By the solution of these problems, a definitive clarification of the foundations of analysis and arithmetic will be obtained, and the system of axioms will be so complete that all remaining questions can be decided by purely logical deduction from them.”

(From Hilbert’s 1900 address, as translated in various editions, e.g., Bulletin of the AMS, 1902). The term “complete” here doesn’t mean Gödel-style completeness yet (that formal definition came later), but Hilbert’s intent was a system where no mathematical truth escapes the axioms’ reach. In his 1922 lecture “The Logical Foundations of Mathematics” (published in From Kant to Hilbert, ed. Ewald), he’s more explicit:

“The chief requirement of the theory of axioms must be that it makes it possible to decide, by means of a finite number of logical inferences, whether any given statement formulated in the language of the theory is true or false.”

This is Hilbert pushing for what we’d now call decidability — every statement provable or disprovable within the system—which implies completeness. He wanted math to be a closed, self-sufficient machine.

Consistency Provable from Within

Hilbert’s demand that consistency be provable “from within” the system is trickier—he didn’t phrase it exactly that way, but it’s how his program was framed, and Gödel’s response hinges on this interpretation. In his 1925 paper “On the Infinite” (Über das Unendliche, Mathematische Annalen), Hilbert outlines his method, later dubbed “finitism,” to secure mathematics:

“We must establish the consistency of the axioms by means of a careful description and analysis of the methods actually used in mathematics… The goal is to establish certainty by finitary means.”

He proposed proving consistency using only “finitary” reasoning — simple, concrete logic he believed was unassailable. Crucially, in his 1928 book Grundzüge der theoretischen Logik (with Wilhelm Ackermann), he posed the “consistency problem” as:

“To prove by finite methods that the axioms do not lead to a contradiction.”

Hilbert didn’t say “from within” verbatim, but his finitist approach aimed to use tools internal to the system’s basic logic, avoiding infinite or external assumptions. He saw this as a proof that the system (e.g., for arithmetic) could vouch for itself. At the 1928 International Congress, he reiterated:

“The development of mathematical science must proceed in such a way that… its freedom from contradiction is proved by methods inherent in the nature of the science itself.”

How Gödel Saw It

Gödel’s 1931 paper (On Formally Undecidable Propositions, Monatshefte für Mathematik) directly engages Hilbert’s goals. He notes that Hilbert’s Program seeks a system where “all true mathematical statements can be derived” (completeness) and where “the freedom from contradiction can be proved by finitary means” (consistency). Gödel then shows this is impossible for arithmetic, citing Hilbert’s own framework as the target.

Why “From Within”?

Hilbert’s insistence on finitary methods — logic he thought was intrinsic to math — implied a self-contained proof. Later scholars (e.g., von Neumann, in letters from the 1930s, and modern texts like Gödel’s Theorem: An Incomplete Guide by Torkel Franzén) interpret this as “within the system” because Hilbert rejected leaning on stronger, external systems, which would undermine his foundationalist aim.

The Evidence

Completeness: 1900 address, 1922 lecture—Hilbert wants all truths decidable from axioms.
Consistency from Within: 1925 paper, 1928 book/talk—proof via finitary, internal means.

Hilbert’s words don’t always use modern terms, but his intent, as read by peers and successors, aligns with these demands. Gödel’s theorems hit precisely because they showed these twin pillars — total coverage and self-proved safety — couldn’t stand together. I can’t paste a single smoking-gun quote, but the pattern across these works is what mathematicians like Gödel and historians like Constance Reid (Hilbert, 1970) point to as the program’s heart.

Okay, I guess Hilbert was wrong, but people wrongfully conclude that mathematical axiomatization does not work. In fact, mathematics can be axiomatized as ZFC, where ZFC is consistent and powerful enough to prove all the theorems of math. It just isn't strong enough to decide every question, or to prove consistency within ZFC.

It was not wrong to try to axiomatize mathematics, not wrong to demand finitary reasoning, and not wrong to want consistency. If he really wanted to prove consistency from within the system, that is not possible, and is not really desirable. Hilbert's real error was to think it is desirable, if he really thought that.

The reason is that an inconsistent system can prove anything. If a system has a contradiction then it can prove 1=2 or anything else. So it does not make any sense for a system to prove itself consistent. It is like asking someone if he is honest. A liar will say that he is honest, and you learn nothing. To assess someone's honesty, you have to ask someone else.

If it is true that Hilbert wanted a math system to prove itself consistent, then he was wrong because that is a nonsensical thing to want. That should have been obvious long before Goedel.

People hear all this and figure that our axiom systems are not strong enough to prove what we want. But actually, Harvey Friedman's grand conjecture says that all our big math theorems can be proved in much weak axiom systems than ZFC, which is what is usually used.