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Concept

Cardinality

The size of a collection, measured by whether its members can be matched one to one with another collection's. It is compared by matching rather than by counting, which is what makes it usable for collections with no last member.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The diagonal, and the row built to be off the list. A table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry.

The row that is not on the list

Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.

logic · Diagonalisation A closed interval and an open one, matched point for point. Two number lines, one closed and one open, with arrows showing the countable sequence of points that has to move.

Two injections make a bijection

If each of two collections fits inside the other without collisions, they are the same size. That sounds obvious and is not, because neither injection needs to be onto — and the proof is a rule for deciding which of the two to follow, one chain at a time.

logic · Cardinality Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

Almost none of it left, and still uncountably many

Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

analysis · Measure A square's worth of points, on a line. A unit square with a point marked, the decimal places of its two coordinates woven into one number, and that number marked on a line beneath.

A line with as many points as a square

Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.

logic · Cardinality The tower of sizes, and the gap in it. A tower of infinite sizes, each the number of sub-collections of the one below, with the space between the first two marked as the one no proof decides.

The size that cannot be pinned down

There is no largest infinity, because no collection has as many members as it has sub-collections. What is not settled is whether anything sits between the first two — and that is not an open problem but a proved absence of an answer.

logic · Cardinality Ordinal sums and products, in normal form. A table of ordinal expressions with their Cantor normal forms and whether the two sides of each pair are equal, above two tick lines drawing one such pair.

One step in front of infinitely many

Put one step before an infinite run of them and nothing has changed; put it after and something has. Ordinal addition records that difference, which is why it is not commutative — and why it keeps information that counting throws away.

logic · Ordinals Limit ordinals and the sequences that approach them. Several ordinals with the first terms of their fundamental sequences, and the successors marked as having a predecessor instead.

Reached from below, or not at all

Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.

logic · Ordinals A countable structure grown by adding witnesses. Stages of a structure built from 0 and 1 by adding sums, products, negatives and roots of quadratics: 2, 4, 12, 158 elements between −3 and 3.

A countable field that passes for the line

The real numbers are uncountable, and every first-order sentence about their addition, multiplication and order is also true of a countable field inside them — the real algebraic numbers. Löwenheim and Skolem showed this is no quirk of the reals: every theory with an infinite model has a countable one, including set theory, which then contains sets it calls uncountable.

logic · Models

Named alongside it

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

BijectionCountabilityAxiomatic set theoryContinuum hypothesisLimit ordinalModelOrder typeOrdinalPower setWell-orderingAlephAlgebraic number

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