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Level: reference · for finding a page by subject instead of by chapter

One line: The sidebar lists the chapters A to Z; this page sorts every lesson by subject into branches, then follows the threads that run between chapters.

The chapter numbers are the suggested reading order, and each chapter is one argument, so inside a chapter the lessons are best read in order. This page is the other way in, for when you know what you want and not which chapter holds it. A lesson sits once in the tree, where it belongs; a ↪ marks a lesson from another branch that is worth reading alongside. The build checks that every lesson appears here.

The tree

  • Numbers
  • Sets and size
    • Building sets · from Sets
      • What is a set? — the definition in four rules: undefined words, extensionality, a set is one object, separation, one empty set
      • The textbook definition on trial — well-defined, "collection" and "distinct", the box picture, null set, Russell's paradox, separation as the repair, no set of all sets, many-valued logics
      • The algebra of sets — universal set as the Venn rectangle, complement (A′, Ā, Aᶜ), disjoint sets, De Morgan's laws, distributive and absorption laws, Boolean algebra, Venn regions, ⊆ as a partial order (poset); each law on its own page: identity, domination, idempotent, complement, double complement, commutative and associative, distributive, absorption, De Morgan, difference
      • Reading a set expression — order of operations for sets, precedence from logic, where difference goes in Python, Lean and Isabelle, Python's - & ^ | ladder, x ∈ A ∩ x ∈ B and x ∈ A ∧ B as bugs, (1, 2) as pair or interval
      • Sets in Python — the bridge: extensionality is ==, separation is a comprehension, | & - ^ as or, and, and-not, xor, no complement without U; links to the Python, Rust and ABAP pages
      • The Cartesian product — ordered pairs, ℝ², and why A × B is not B × A
      • Relations and functions are sets of pairs — relation, domain, range, inverse, function as a relation with one property, injection, surjection, bijection, composition as a relation product, graph as a relation drawn, database table
      • Equivalence relations, partitions and the kernel of a function — reflexive, symmetric, transitive; classes, quotient, Bell numbers, kernel, decomposition of a function, congruence mod m; finer and coarser, the lattice of equivalences, the equivalence generated by a relation (transitive closure, connected components), well-definedness on classes, orbits of a group action and Burnside's count
      • Multisets — multiplicity, Counter, min and max as ∩ and ∪, sum, gcd and lcm as operations on factorisations, stars and bars, no complement
      • Ramsey's theorem — pigeonhole for pairs, R(3, 3) = 6 by exhaustion, the infinite theorem as an algorithm, König's lemma, Erdős–Szekeres, arrow notation, weakly compact cardinals: where combinatorics becomes set theory
      • Orderings — reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, comparable; equivalence, partial order, poset, linear order, strict order; chain, antichain, maximal, minimal, maximum, minimum; lexicographic order; Riemann partition and refinement
      • Set katas — Cunningham's exercises 1.1 checked on every choice of subsets of {1, 2, 3}; element chase, unpacking a negation, two inclusions, truth sets, intervals; ten quiz questions on subsets, proper subsets, factors and multiples, with divisibility as ⊆; four pasted precedence katas on Python's - & ^ | ladder, tried under every bracketing
      • Function katas — Hrbacek and Jech 3.1–3.13 and 4.1–4.3 with solutions: domain of a composition, inverses, left and right inverses, (g ∘ f)⁻¹, images and preimages, reflexive–symmetric–transitive table
      • ↪ Set theory: a reading guide — the language of sets and the subject of its axioms, which book for each, the Polish school, and the founding papers from Cantor 1874 to Cohen 1963
      • ↪ Set theory: the terms — the owner's list of some 700 terms, each linked to the lesson or glossary entry that covers it, the rest named with the book that does
      • ↪ Rectangular coordinates — the pairs of ℝ² as points of the plane
    • Size by matching · from Sets and Measure Zero
      • Cardinality of sets — size defined by matching, not counting
      • Cantor–Schröder–Bernstein — injections both ways give a bijection; ancestor chains; ≤ on cardinals is a partial order without choice; [0, 1] and (0, 1)
      • ↪ Power set — Cantor's theorem, |𝒫(a)| > |a|, as an axiom's consequence
      • ↪ Set theory: a reading guide — Cantor 1874, Dedekind 1888 and Cantor 1891 run as a program: listing the algebraic numbers, infinite as matching a proper part, the diagonal
      • Countable sets — any set that can be listed, even the rationals, has measure zero
    • The axioms of set theory · from Axioms of Set Theory
      • Reading a formula — ∀ ∃ ⇒ ⇔ ≃ ∈ v₀ v₁ as loops and lookups, the universe as a dictionary, every axiom evaluated on V₂, V₃, V₄ and a universe with atoms
      • Extensionality — same members, same set; a ⊆ b and b ⊆ a; "unique by extensionality"; urelements
      • Pairs — {a, b}, {a}, rank, Kuratowski's ordered pair from three pairs, pairing as a theorem of replacement over 𝒫(𝒫(∅))
      • Unions — ∪a versus a ∪ b, rank goes down, ∩∅ does not exist
      • Power set — ⊆ inside the formula, V_{n+1} = 𝒫(V_n), Cantor's theorem
      • Comprehension — a scheme, {x ∈ a : F[x]}, ∅ ∩ ∖ derived, classes, Russell's paradox as a truth table
      • Replacement — functional formulas, images, comprehension for free, V_{ω+ω} and the set Zermelo cannot collect
      • Infinity — successor, inductive set, ω as the intersection, no finite universe has one, limit ordinal
      • Choice — choice function, product of nonempty sets, finite families, shoes and socks, Zorn, Teichmüller, Kurepa, well-ordering, the prime ideal theorem, Banach–Tarski, Gödel and Cohen
      • Two balls from one — the free group doubles itself, Hausdorff's rotations checked free in exact arithmetic, the labelled Cayley graph, where the axiom of choice enters, why the pieces have no volume
      • Foundation — ∈-minimal member, no x ∈ x, rank, Quine atoms, the two-cycle and {a, b}, anti-foundation
      • Ordinals — transitive and well-ordered by ∈, the von Neumann numbers, Goodstein sequences, Kirby and Paris
      • Axiom katas — Jech 1.1–1.15, Cunningham 1.4 and 1.5, Kunen I.2.1, checked on a small universe; the five proof moves
      • ↪ What is a set? — extensionality, separation and the empty set in plain words, before the formal language
      • ↪ The textbook definition on trial — Russell's paradox and the repair before the formal language
      • ↪ The Cartesian product — Kuratowski's pair put to work
      • ↪ Set theory: a reading guide — Cori and Lascar, Cunningham, Jech and Kunen placed among the books
      • ↪ Lean: a reading guide — Mathlib's Set α := α → Prop run on a four-element type, the laws of set algebra proved by extensionality the way Set.ext does
      • ↪ Proof assistants: a reading guide — a proof checker with Isabelle's rule names, model check against proof, the ZF axioms as Isabelle/ZF states them, Paulson's manual placed, Python, Rust, Lean and Isabelle for sets
    • Size by length · from Measure Zero
  • Algebra
  • Geometry
  • Calculus
  • Analysis
    • Metric spaces: limits from a distance · from Metric Spaces
      • Open balls: three metrics, the same open sets — An open ball is the points closer than r; a set is open when every point keeps a ball inside; cheb ≤ eucl ≤ taxi ≤ 2 cheb proved in three lines and checked on 2 401 pairs, so the three balls nest and the three metrics share every open set; the discrete metric, whose balls are a point or everything, does not
      • Convergence: a limit is a statement about every ball — x_n → L when every ball around L holds a tail; N found for 1/n and three radii with the algebra that makes it work for all n; (−1)ⁿ has no limit by one ball; limits are unique by two disjoint balls; three equivalent metrics give three N and one limit; under the discrete metric 1/n stops converging
      • Continuity by ε and δ — f sends a δ-ball around p into the ε-ball around f(p); δ = min(1, ε/7) for x² at 3 with the two-line reason; the step function has one ε that defeats every δ, witness x = −δ/2; the distance to a point is continuous with δ = ε by the triangle inequality; the identity between equivalent metrics is a homeomorphism; out of a discrete space every function is continuous
      • Cauchy sequences and completeness — Terms within ε of each other, no limit named; Newton's iteration for √2 in exact fractions is Cauchy in ℚ and no rational squares to 2, so ℚ has holes; ℝ is complete by construction, √2 to 12 decimals by isqrt; (0, 1) and ℝ share their open sets and differ in completeness, so completeness is not topological; the discrete metric and ℤ are complete cheaply
      • The triangle inequality proved: Cauchy–Schwarz — |u + tw|² is a quadratic in t that is never negative, so its discriminant is ≤ 0, which is (u·w)² ≤ |u|²|w|²; equality exactly for parallel vectors; checked on 15 625 pairs; the triangle inequality for the Euclidean distance in three lines without a square root, paying the debt the chapter ran up
      • ↪ Distance is the four properties — the four axioms and the three circles whose insides are the balls
      • ↪ Distance in n dimensions — the Euclidean metric in any number of coordinates, its triangle inequality left open there and proved here
      • ↪ Predicates and quantifiers — ∀ε ∃δ, and what negating it says
      • ↪ What measure zero means — analysis on ℝ, which this chapter's completeness page says ℝ was built for
      • ↪ The derivative is a velocity — the limit that chapter used before it was defined
  • Logic and proof
    • If A then B: converse, contrapositive, if and only if · from Logic
      • If A then B: converse, contrapositive and inverse — implication, hypothesis, conclusion, truth table, converse, inverse, contrapositive, if and only if, counterexample, affirming the consequent
      • Truth tables and the laws of logic — ¬ ∧ ∨ → ↔, tautology, contradiction, logical equivalence, ↔ versus ⇔, De Morgan, distributive, contrapositive, → does not associate, a misprint caught by one row
      • Predicates and quantifiers — predicate, universe of discourse, ∀ ∃ as loops, bounded quantifiers, interchange, negation flips a quantifier, distribution, ∃!
      • What a proof is — givens, claim, chain; a reason beside every line; letters cover every case where a loop cannot; proof by cases and the check that the cases are exhaustive; a circle, assuming the claim, a wrong line with a good shape; a number check refutes and never confirms; the distance formula as ten lines
      • Induction — base case, induction hypothesis, inductive step, well-ordering principle, least counterexample, proof by contradiction, the all-cars proof, Fibonacci, Binet's formula, 2ⁿ subsets, chocolate bar, invariant, closed form, sigma notation
      • ↪ Lean: a reading guide — a proposition is a type and a proof is a program: And is a pair, Or a tagged value, ¬A a function into False; a type checker for proof terms, and Mathlib's sets as predicates; the free books and Lean Beam
      • ↪ Proof assistants: a reading guide — natural deduction checked by program: conjI, impI, mp, notI and the one classical rule; what a proof covers that a truth table does not
      • ↪ Infinity — the inductive set that makes induction a sentence about ℕ
      • ↪ Ordinals — well-ordering past ℕ, and a theorem induction on ℕ cannot prove
      • ↪ Reading a formula — the same symbols inside the axioms of set theory, evaluated on a small universe
      • ↪ The Pythagorean theorem and its converse — a theorem whose converse is also true
      • ↪ Distance is the four properties — one triple of points disproves a metric; four properties checked on all 512 triples of bit strings
      • ↪ A definition is a test — a definition as an "if and only if"
      • ↪ The textbook definition on trial — Russell's paradox: an argument by cases in which both cases fail
      • ↪ The algebra of sets — ∪ ∩ ′ are or, and, not; ⊆ is if–then; De Morgan's laws
      • ↪ Day or night: perspective taking ↗ — a definition by cutoff, the sorites, and fuzzy logic's "day and not day"
  • Chance and data
  • Learning to learn, now the Learning to Learn library ↗; kept here as the map of where each lesson went

Threads across chapters

Each thread follows one idea through lessons in different chapters. They are the links the lessons already make to one another, gathered in one place.

From a set to its axioms

The same objects three times: in everyday use, in Python, and on the axioms.

  1. What is a set? → The textbook definition on trial — the definition in four rules, then the textbook's sentence, the rules it leaves out, and Russell's paradox.
  2. The algebra of sets → Truth tables and the laws of logic — ∪ ∩ ′ are ∨ ∧ ¬, so one finite table proves both lists of laws.
  3. Predicates and quantifiers → Reading a formula — ∀ and ∃ as loops, first over four people, then over a universe of sets.
  4. The Cartesian product → Pairs → Relations and functions — Kuratowski's pair, the axiom that makes it, and everything built from pairs.
  5. Cardinality of sets → Power set → Infinity → Ordinals — size by matching, then the axioms that give an uncountable set, an infinite one, and the ordinals that count past it.
  6. Set theory: a reading guide — the founding papers of steps 1 to 5, four of them run as a program, and the books to continue in.
  7. Proof assistants: a reading guide — the same axioms as Isabelle/ZF states them, and a proof where steps 1 to 5 had model checks.

Floats and exact numbers

A computer holds a finite set of numbers, and that fact reaches well beyond chapter 1.

  1. Machine numbers — the finite set, and which laws of arithmetic survive rounding into it.
  2. The laws of an operation and A definition is a test — floats fail associativity and distributivity, because each grouping rounds at a different step.
  3. Multiplication as pairs — Python's complex is two floats, so every warning applies twice.
  4. Probability zero — a finite set has measure zero, so a real number picked at random is almost never a float.
  5. Exact vs approximate → Mean, average, arithmetic mean — a count is exact, so dividing by it costs no significant figures.
  6. Linear equations and their solutions — exact fractions, so "a tuple passes" means the two sides are equal, not nearly equal.
  7. The derivative is a velocity — a derivative measured in floats cannot shrink its step for ever: past h = 10⁻⁸ cancellation takes over.
  8. Euler's identity — cmath.exp(1j * math.pi) is not −1: the double nearest π falls short of it by 1.2 × 10⁻¹⁶, and the answer sits exactly that far above the axis.

Two kinds of size

Matching and length are different questions about how big a set is, and they can give opposite answers.

  1. Cardinality of sets — size by matching: ℕ is the same size as its even numbers, and ℝ is larger.
  2. The Cartesian product → Countable sets — a product of finite sets can be listed, and ℝ × ℝ cannot.
  3. Countable sets — the matching of ℕ with the rationals, which makes them measure zero.
  4. The Cantor set — uncountable, as large as ℝ by matching, and still length 0.

Matrices in disguise

Several lessons meet 2 × 2 matrices before any chapter is about them.

  1. Multiplication as pairs and Multiplication rotates → Maps that keep the laws — the pair (x, y) behaves exactly like the matrix [[x, −y], [y, x]], so building ℂ from pairs and building it from matrices build the same thing.
  2. Multiplication can be undone → The laws of an operation — zero divisors, and why the 2 × 2 matrices are a ring and not a field.
  3. Linear equations and their solutions — the coefficients aᵢ,ⱼ of a system, laid out in rows and columns, are where matrices come from.

Definitions as tests

A definition in the axiomatic style is a test that objects pass or fail, and a theorem proved from the test holds for everything that passes.

  1. A definition is a test — the vector-space axioms as a test for whole sets.
  2. Subsets inherit the laws — the shorter test for a subspace.
  3. Linear equations and their solutions — the same idea one level down: an equation is a test for a tuple.
  4. If A then B: converse, contrapositive and inverse — a definition is an "if and only if" by agreement, while a theorem's converse needs its own proof.

The plane as pairs

ℝ² is a set of ordered pairs, and three chapters use it as a plane.

  1. The Cartesian product — the set: every (x, y) with x and y real, and why the order matters.
  2. Rectangular coordinates — the plane: each pair is a point, each coordinate a signed distance from an axis, and the four quadrants are the four pairs of signs.
  3. Multiplication as pairs → Multiplication rotates — the same points with a multiplication, which turns them; multiplying by i moves a point one quadrant counterclockwise.
  4. Linear equations and their solutions — one equation in x and y has a line of solutions, and each solution is a point of the plane.
  5. A definition is a test — the same pairs read as vectors, displacements rather than locations, and the axioms they pass.

Pythagoras everywhere

One theorem about right triangles turns out to be how every distance in this library is measured.

  1. The Pythagorean theorem and its converse — c² = a² + b², and the converse that turns it into a test.
  2. Area and volume formulas — the diagonal of a square, and the altitude of a triangle, both found by it.
  3. Congruent and similar triangles — why two sides of a right triangle fix the third, the one case where SSA works.
  4. The distance formula — the theorem with the legs read off coordinates.
  5. What a proof is — the distance formula as a ten-line chain, in which the right angle costs one named assumption.
  6. Circles: standard form and general form — the distance formula held fixed, with twelve whole-number points on x² + y² = 25.
  7. Distance in n dimensions, and the length of a vector — the theorem applied twice for a box, n − 1 times for n axes, and the norm √(v·v) as its distance from the origin.
  8. Distance is the four properties — the theorem's distance beside two others, and the rotation test that singles it out.
  9. The triangle inequality proved: Cauchy–Schwarz — the fourth metric property of the theorem's distance, from a quadratic with no roots.
  10. Lines: slope, equations, parallel and perpendicular — perpendicular slopes, checked by the converse.
  11. Multiplication rotates — the length of a complex number, √(x² + y²), is the same formula measured from 0.
  12. Catastrophic cancellation — the Burj Khalifa example subtracts two nearly equal squares, and a calculator pays for it.
  13. Related rates — ladders and travellers: x² + y² = z², differentiated in time.
  14. Studying vs learning ↗ — Bloom's six levels climbed on the theorem, up to Euclid's formula for every whole-number right triangle.

Scale factor k

  1. Area and volume formulas — lengths times k, areas times k², volumes times k³.
  2. Congruent and similar triangles — similar triangles are one triangle scaled by k.
  3. Multiplication rotates — multiplying by a complex number scales by its length and turns, so every triangle goes to a similar one.
  4. The midpoint formula — k = ½: halfway across and halfway up is halfway along.
  5. Lines: slope, equations, parallel and perpendicular — every rise-over-run triangle on a line is a scaled copy of every other.
  6. Related rates — the cyclists' triangle grows by a scale factor 20t, so its rate of growth never changes.

From exponents to Euler's identity

The path of Euler's formula: a lesson plan, through the pages this library has for it.

  1. A definition is a test → Maps that keep the laws — the exponent laws, and the exponential as the map that turns adding into multiplying.
  2. Rectangular coordinates → Multiplication as pairs — the plane as pairs, and a complex number as a point of it.
  3. Multiplication rotates — multiplying by i is a quarter turn.
  4. Radians → Roots of unity — the unit circle measured by distance walked, and twelve points of it computed exactly.
  5. The derivative is a velocity → Velocity equals position — the talk's two arrows, and the rule velocity = k · position.
  6. Euler's identity — k = i, and the point goes round the circle to −1.
  7. Power series — the other road, adding arrows instead of moving.

The complex numbers as a field

  1. Multiplication as pairs — the pairs, with their rule, form a field.
  2. A definition is a test — in any field, a polynomial of degree n has at most n roots.
  3. Roots of unity — which is how the lesson knows that the n solutions of zⁿ = 1 it finds are all of them.

Auf Deutsch: Stichwörter

Jede Lektion nach Fachgebiet sortiert: ein Baum mit Zweigen, Querverweisen und Fäden, die eine Idee durch mehrere Kapitel verfolgen.

Stichwörter: Themenkarte, Fachgebiet (subject), Zweig (branch), Querverweis (↪), Faden (thread), Lesepfad.