Hilbert's Questions

David Hilbert · 1862–1943

Who he was

David Hilbert — the man who mapped the unknown.

David Hilbert (1862–1943) was one of the most influential mathematicians who ever lived — his work reshaped geometry, algebra, logic, and the very foundations of mathematics.

In 1900, at the International Congress in Paris, he did something bold: he stood up and laid out a list of the great unsolved problems — a map of the unknown to guide mathematics through the coming century.

Those questions weren't puzzles for their own sake. They reach into the deep structure of number, space, and infinity — and answering one tends to open far more than math.

InfinityThe continuumFoundationsThe shape of number

The questions he asked

All 23 problems.

Hilbert posed 23 problems in 1900 — his map of the unknown. Here they all are, in plain language. We're working through them one at a time in the Key's own mathematics, and posting each answer in the section below.

So far we lead with his First — the Continuum (worked) — and his Second is seeded.

The Continuum

Is there a size of infinity between the counting numbers and the real line?

Worked

Consistency of arithmetic

Can arithmetic be shown never to contradict itself?

Seeded

III

Equal tetrahedra

Do two tetrahedra of equal base and height have equal volume by cutting?

Straight-line geometries

Characterize all geometries in which lines are the shortest paths.

Continuous groups

Must a continuous transformation group automatically be differentiable?

Axioms of physics

Give physics a rigorous, axiomatic mathematical foundation.

VII

Transcendence

Is aᵇ transcendental for algebraic a and irrational algebraic b?

VIII

The Riemann Hypothesis

The hidden pattern of the primes — with Goldbach and the twin primes. (Also a Millennium question.)

Reciprocity laws

Find the most general law of reciprocity in a number field.

Diophantine decision

Is there a single method to decide any equation in whole numbers?

Quadratic forms

Solve quadratic forms with algebraic number coefficients.

XII

Kronecker's dream

Extend the theory of abelian fields to any base field.

XIII

Seventh-degree equations

Can the general 7th-degree equation be solved using functions of two variables?

XIV

Finiteness of invariants

Are certain complete systems of functions finitely generated?

Schubert's calculus

Put enumerative geometry on a rigorous footing.

XVI

Curves and cycles

The topology of real algebraic curves and surfaces; limit cycles.

XVII

Sums of squares

Is every positive-definite rational function a sum of squares?

XVIII

Filling space

Tiling and sphere-packing with congruent shapes.

XIX

Analytic solutions

Are solutions of regular variational problems always analytic?

Boundary-value problems

Do all well-posed boundary problems have solutions?

XXI

Prescribed monodromy

Do linear differential equations with a given monodromy exist? (Riemann–Hilbert.)

XXII

Uniformization

Uniformize analytic relations by automorphic functions.

XXIII

Calculus of variations

Develop the methods of the calculus of variations further.

We are turning the Key on them one at a time — each answer will be posted below, with its document.

XXIV

The lost 24th · the most profound, for us

And then there's the one he left out.

Hilbert drafted a 24th problem — and then quietly left it off his famous list. It stayed hidden in his notebooks for a century, until a historian found it in his notes in 2000.

What it asks is unlike the other twenty-three. Not a problem about number or space, but about mathematics itself: what makes a proof the simplest it can be? A criterion for simplicity — a way to know when you've reached the plainest, most direct path to a truth.

For Trinary, this isn't one question among twenty-four. It's the whole idea. The 28 is built from the simplest things there are — a point, a line, a circle, a fold. The Key's whole wager is that the simplest form is the true one: you reach an answer by returning to the source, not by piling on machinery.

Hilbert's lost question is the one our entire method is an answer to.

The answers

What the Key turned out.

Each answer we work gets posted here — a plain-language write-up and the full worked document to download. This list grows over time.

Worked2026

The Continuum — Hilbert's First

We turned the Key on the Continuum. On the Key's own foundation, the continuum is the immediate doubling-fold, with no size of infinity sitting between the counting numbers and the real line — so CH holds at the source. This is the Key's own result, not a proof inside standard set theory (ZFC's independence still stands); the full working, every check, and the standard view set beside it are in the document. What came out alongside it: a new set theory and a new number theory, from the same small object.

ViewWorked answer — posting shortlyPDF · the full working

In progress

The consistency of arithmetic — Hilbert's Second

Seeded, not yet opened. The question Hilbert's own program broke on, exactly where Gödel's second theorem bites. The Key's closing fixed point offers a positive self-consistency certificate on the branch Gödel's result doesn't govern — held honestly, to honor Gödel and fill the gap, never to contest it. The write-up and document post here when it's worked.

ViewDocument coming soonposts here when ready

In progress

A rule for Diophantine equations — Hilbert's Tenth

In progress. Hilbert asked for a single procedure that could decide, for any whole-number polynomial equation, whether it has a solution. Standard mathematics settled the general case in 1970 (Matiyasevich, building on Robinson, Davis and Putnam): no such universal procedure exists. Where the Key looks is that boundary itself — what the source says about which equations close and which stay open. Held to the same honesty: the standard result stands; this is a different lens, not a contest. The write-up and document post here when it's worked.

ViewDocument coming soonposts here when ready

David Hilbert — the man who mapped the unknown.

All 23 problems.

And then there's the one he left out.

What the Key turned out.

The Continuum — Hilbert's First

The consistency of arithmetic — Hilbert's Second

A rule for Diophantine equations — Hilbert's Tenth

Have a question you'd turn the Key on?