Generator

infinite

A generator in the logic library, called 9 times across 4 essays. Below: what it draws at its defaults 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.

At its defaults

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

`show: "interleave"`

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.

`show: "height"`

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.

`show: "zigzag"`

`show: "tower"`

The tower of sizes, and the gap in it. A ladder 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.

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
every fraction the grid reaches is on the list ×1
every kept cell got exactly one place in the list ×1
no fraction is listed twice ×1
the complemented diagonal is on no row of the listing ×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 places run from one with no gaps ×1
the second number has the same number of places ×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

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

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

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

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.

The whole library · What the figures prove