Learn Theoremsv1.0

Liouville's Theorem (Cauchy/Liouville)

An entire function that is bounded must be constant; global constraints on growth force a function's derivative to vanish. Step through the replay to see …

math.CV · Complex Variables · math.AP

📚 Undergraduate · Complex Analysis

Lebesgue Number Lemma (Henri Lebesgue)

In any compact metric space, every open cover admits a Lebesgue number: a positive constant such that any subset of smaller diameter must be contained within a…

math.GN · General Topology · math.MG

📚 Undergraduate · Topology

Lagrange's Theorem (subgroup order divides group order)

If H is a subgroup of a finite group G, then |H| divides |G|. The first structural theorem of group theory; its proof partitions G into equal-sized cosets. Ste…

math.GR · Group Theory

📚 Undergraduate · Algebra

Krull's Intersection Theorem (Wolfgang Krull)

In a Noetherian local ring, the intersection of the powers of a proper ideal annihilates the intersection itself. Step through the proof; see how the Artin-Ree…

math.RA · Rings and Algebras · math.AC · math.AG

📚 Graduate · Algebra

König's Theorem (König)

König's theorem proves that the sum of a sequence of cardinals is strictly less than their product, provided each component satisfies the strict inequalit…

math.LO · Logic · math.GM

📚 Undergraduate · Set Theory

König's Lemma (Dénes Kőnig)

Every infinite, finitely branching tree contains an infinite branch. This fundamental result bridges finite and infinite combinatorics. Step through the replay…

math.LO · Logic · math.CO · math.GM

📚 Graduate · Set Theory

Kleene's Normal Form Theorem (Kleene)

Every partial computable function is expressible as a primitive recursive extraction from the result of a minimal search over a primitive recursive computation…

math.LO · Logic · math.GM

📚 Graduate · Logic

Jordan–Brouwer Separation Theorem

An embedding of S^{n-1} in R^n divides the space into exactly two connected components: one bounded and one unbounded. Step through the replay to see how the w…

math.AT · Algebraic Topology · math.DG

📚 Graduate · Topology

Intermediate Value Theorem (Bolzano)

For a continuous function f on [a, b], any value u between f(a) and f(b) is attained by some c in (a, b). Continuity prevents the function from skipping values…

math.CA · Classical Analysis and ODEs · math.RA

📚 Undergraduate · Real Analysis

Hurewicz Theorem (Hurewicz)

The Hurewicz homomorphism identifies homotopy and homology in the first dimension where they are non-zero. The theorem bridges discrete homotopy with abelianiz…

math.AT · Algebraic Topology

📚 Graduate · Topology

Homotopy Lifting Property (Hurewicz, Eilenberg, Steenrod)

Covering spaces are defined by their path lifting, but they also extend homotopies. The Homotopy Lifting Property ensures any homotopy of a map into the base s…

math.AT · Algebraic Topology · math.GT

📚 Graduate · Topology

Hilbert's Nullstellensatz (Hilbert)

The Nullstellensatz establishes a perfect dictionary between affine varieties and radical ideals over an algebraically closed field. The proof relies on the in…

math.RA · Rings and Algebras · math.AG · math.AC

📚 Graduate · Algebra

Heine–Borel Theorem (Heine, Borel)

A subset of Euclidean space is compact if and only if it is closed and bounded. The proof uses the nested interval property to find a point that contradicts th…

math.CA · Classical Analysis and ODEs · math.FA · math.MG

📚 Undergraduate · Real Analysis

Hartogs' Lemma (Hartogs)

For every set A, there exists an ordinal \alpha such that no injection from \alpha into A exists. This demonstrates that the class of ordinals is unbounded in …

math.LO · Logic · math.GM

📚 Graduate · Set Theory

Hahn–Banach Theorem (Hahn, Banach)

A bounded linear functional on a subspace can be extended to the whole space while preserving its norm, or more generally, its domination by a sublinear functi…

math.FA · Functional Analysis · math.LO

📚 Graduate · Real Analysis

Gödel's Second Incompleteness Theorem (Gödel)

A consistent, sufficiently strong formal system cannot prove its own consistency statement Con(T). A cornerstone of modern logic; step through this formalizati…

math.LO · Logic · math.GM

📚 Graduate · Logic

Gödel's First Incompleteness Theorem (Gödel)

Any consistent, effectively axiomatized formal system $F$ capable of expressing basic arithmetic contains a sentence $\sigma$ such that neither $\sigma$ nor $\…

math.LO · Logic

📚 Graduate · Logic

Fundamental Theorem of Cyclic Groups

Every subgroup of a cyclic group is cyclic. If a finite cyclic group has order n, for each divisor k of n there exists exactly one subgroup of order k. Step th…

math.GR · Group Theory · math.RA

📚 Undergraduate · Algebra

First Isomorphism Theorem (Noether)

The kernel of a group homomorphism captures the redundancy of the map. By quotienting the domain by the kernel, we recover an isomorphic image in the codomain.…

math.GR · Group Theory · math.RA

📚 Undergraduate · Algebra

Extreme Value Theorem (Weierstrass)

Continuous functions on a closed, bounded interval [a, b] always attain their absolute maximum and minimum. This result bridges the gap between boundedness and…

math.CA · Classical Analysis and ODEs · math.FA

📚 Undergraduate · Real Analysis

Excision Theorem (Eilenberg-Steenrod)

The Excision Theorem allows computing relative homology by removing subspaces that do not contribute to the homology of a pair. It relies on barycentric subdiv…

math.AT · Algebraic Topology · math.GT

📚 Graduate · Topology

Downward Löwenheim–Skolem Theorem

First-order logic cannot fix the cardinality of infinite structures. Any infinite structure has a small elementary substructure that satisfies the same sentenc…

math.LO · Logic · math.GM

📚 Graduate · Logic

The Deduction Theorem (Herbrand/Tarski)

A fundamental metatheorem bridging hypothetical reasoning with unconditional proof. It converts a derivation of B from hypothesis A into a formal proof of A → …

math.LO · Logic · math.GM

📚 Undergraduate · Logic

Completeness Theorem for Propositional Logic (Post)

If a formula is semantically valid (true in all models), it is formally provable. This bridge between truth and syntax defines logic. Step through the replay t…

math.LO · Logic · math.GM

📚 Undergraduate · Logic