Concept

Hilbert hotel

The illustration in which a hotel with a room for every whole number is full and can still take another guest. It makes the shifting bijection vivid and it is not itself the argument, since the correspondence is a single function rather than a sequence of moves in time.

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

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 $The fractions, put in a line. A grid whose rows are numerators and columns denominators, walked by antidiagonals, with the place each fraction takes in the list written in its cell and the repeats left blank.$

The arithmetic that loses subtraction

Adding one to an infinite collection changes nothing, and neither does doubling it, or squaring it. What that costs is the two operations that were doing the work — an equation between infinite sizes cannot be cancelled, and how many are left stops being a question.

logic · cardinality

Named alongside it

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

Bijection Countability Absorption Cancellation Cardinal arithmetic Cardinality Chain decomposition Construction Infinite set Injection Interval Pairing function

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