Matpic
mathematics, in pictures
Concept

Non-measurable set

A set of points to which no length, area or volume can be assigned consistently with translation and countable addition. Such sets exist because of the axiom of choice, and a ball can be cut into five of them and reassembled into two balls.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Every way of choosing one thing from each of 4 pairs. A table with one row per choice function on a small family of pairs, each row giving what it takes from each pair, with the row a stated rule names picked out.

The choice nobody can write down

Given finitely many pairs, picking one thing from each is a finite list of decisions and needs no justification. Given infinitely many, the list cannot be finished — and whether one exists anyway is an axiom, independent of everything else, whose consequences include a theorem most people refuse to believe.

logic · Axiom of choice The share of a ball that is its own edge. A plot of the proportion of each ball formed by its outermost shell against the radius, one line per group — falling towards nothing for the lattice groups and holding steady for the free group.

The edge that is as big as the ball

In a lattice the boundary of a large ball is a negligible fraction of it. In a tree it is two thirds of it at every size — and that single ratio, not the group's size, is what decides whether a set can be cut into pieces and reassembled into two copies of itself.

algebra · Cayley graph An additive function that is nowhere a line. A square window with 1157 points of the graph of an additive function that sends √2 to 0, scattered across the window and meeting every part of it, beside a dashed diagonal marking the straight line y = x.

A function that adds and is nowhere a line

Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.

logic · Axiom of choice Infinitely many guessers, finitely many wrong. Three rows over the first 40 places: the hats worn, the chosen representative of their class, and a row of marks showing each guess right or wrong. The 5 wrong guesses all fall within the first 14 places, up to a marked place; every later guess is right.

Infinitely many guessers, finitely many wrong

An infinite line of people each wears a black or white hat, sees every hat in front and none of their own, and must guess their own colour. With a finite line, each guesser is right half the time whatever they agree in advance. With an infinite line and the axiom of choice, they can agree a strategy under which all but finitely many are right — and nobody can carry it out.

logic · Axiom of choice Three sizes of infinity, and the collections that have each. ℵ₀: the fractions, the algebraic numbers, every finite text, every definition of a number; 2^ℵ₀: the real numbers, the continuous functions, the open sets, the closed sets, the Borel sets; 2^(2^ℵ₀): all functions, all sets of reals, the non-Borel sets, the measurable sets.

What counting can prove exists

There are only as many continuous functions as points of a line, only as many Borel sets, only countably many definitions — and more sets of reals than any of those. So counting proves that discontinuous functions, non-Borel sets and undefinable numbers exist without exhibiting one. It cannot prove that a set with no length exists, because the sets that have a length are exactly as numerous as all sets, and that difference turns out to be the difference between what the axioms force and what they merely allow.

logic · Cardinality A plane split by the order of a list. first 90 rationals; A has 4095 of 8100 pairs; 2/5 is listed at position 7.

The plane cut by a list

The continuum hypothesis looks like a statement about sizes: no infinity sits strictly between the whole numbers and the line. Wacław Sierpiński found in 1919 that it is equally a statement about the plane. It is true exactly when the plane splits into two pieces, one meeting every vertical line in only countably many points and the other meeting every horizontal line in only countably many. The split is drawn by listing the line's points in a particular order, and with the rationals in place of the reals it can be drawn exactly — a set that, counted column by column, has no area, and counted row by row, has all of it.

logic · Cardinality

Named alongside it

The objects these essays reach for when they reach for this one.

Axiom of choiceIndependenceCardinalityChoice functionCountabilityWell-orderingZorns lemmaBasisCantor setCayley graphContinuityContinuum hypothesis

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