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Cardinality — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · logic
  2. 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.

    part 2 · logic
  3. 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.

    part 3 · logic
  4. 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.

    part 4 · logic
  5. 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.

    part 5 · logic
  6. 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.

    part 6 · logic
  7. A countable closed set stripped of its isolated points, 3 times. Cantor–Bendixson derivatives of the set of sums of up to 2 reciprocals: sizes 750, 40, 1, 0 in the truncation, ending empty.

    Closed sets obey the continuum hypothesis

    Whether some set of reals is bigger than the whole numbers and smaller than the line is a question the axioms cannot answer. For closed sets it has an answer, found in 1883: strip away the isolated points, again and again, and what is left is either nothing or a perfect set, which is as large as the line — so every closed set is countable or of the line's size. The same answer holds for Borel and analytic sets, and fails to be provable exactly one step further up.

    part 7 · logic

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