Generator

The fractions, put in a line

A generator in the logic library, called 32 times across 7 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

infinite is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

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

A square's worth of points, on a line

The tower of sizes, and the gap in it

The fractions, put in a line

A countable closed set stripped of its isolated points, 3 times

Splitting a perfect set in two, again and again

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

a set of 0 has more subsets than members ×13
height 2 holds finitely many polynomials, and the enumeration produced them ×5
−1 is among the roots the enumeration found ×1
√2 is among the roots the enumeration found ×1
1 is among the roots the enumeration found ×1
a half is among the roots the enumeration found ×1
a larger height admits more polynomials ×1
all 512 listings of 3 subsets of a set of 3 were tried ×1
and the second ×1
and yet the two images are far apart, so the weave is not continuous ×1
by a factor of more than a thousand at this many places ×1
each length contributes finitely many words ×1
each stage keeps two thirds of the length before it ×1
every fraction the grid reaches is on the list ×1
every kept cell got exactly one place in the list ×1
every piece is a genuine interval ×1
every point claimed to be a limit has points of the set crowding onto it ×1
more fractions pin the curve more tightly ×1
no fraction is listed twice ×1
no word is listed twice ×1
one step before, only the point 0 is left ×1
the complemented diagonal is on no row of the listing ×1
the derivatives reach the empty set ×1
the digit strings are as long as the drawing says ×1
the first number's digits are between four and ten decimal places ×1
the golden ratio is among the roots the enumeration found ×1
the largest height enumerated is a whole number between 3 and 6 ×1
the largest sum of numerator and denominator drawn is a whole number between 3 and 9 ×1
the number of places drawn is a whole number between 4 and 10 ×1
the number of rungs drawn is a whole number between 3 and 5 ×1
the pieces at each level are disjoint ×1
the places run from one with no gaps ×1
the second number has the same number of places ×1
the set is built from one to three reciprocal terms ×1
the size at which every listing is tried is a whole number between 2 and 3 ×1
the two expansions differ by less than one unit in the last place drawn ×1
the view is one the family draws ×1
the walk's count and the sum of totients agree, having been computed independently ×1
the window holds enough of them to draw ×1
the window on the line is between one and twelve wide ×1
the woven number has one place from each, alternately ×1
unweaving recovers the first number ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Logic

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

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.

Logic

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

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

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

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

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.

The whole library · What the figures prove