Concept

Measure zero

The property of a set that can be covered by intervals of total length as small as desired. Every countable set has it, and so do uncountable ones such as the Cantor set — so it is a notion of smallness quite separate from how many members a set has.

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

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

The algebraic numbers, arriving in finite batches. A stretch of the number line with the roots of integer polynomials marked, each at the height of the smallest polynomial that catches it, and the count of polynomials at each height.

Countable, and everywhere

The numbers a polynomial can catch arrive in finite batches, so they can be listed. They are also in every interval, however short. Being listable turns out to say nothing whatever about being sparse.

logic · cardinality

Named alongside it

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

Algebraic number Baire category Bijection Cantor set Cardinality Countability Density Enumeration Fractal dimension Geometric series Height Listing

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