Generator

The 8 subsets of a set of 3, ordered by inclusion

A generator in the logic library, called 42 times across 10 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.

lattice 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 8 subsets of a set of 3, ordered by inclusion. A Hasse diagram of the subsets of a small set, with an edge wherever one subset is another plus one element.

A map from 3 elements into the 8 subsets, and the subset it misses

Everybody's share of the 24 chains

$Everybody's share of the 24 chains. The subsets of a set of 4, each labelled with the fraction of maximal chains it lies on; the shares of any antichain add to at most one, and to exactly one only for a whole layer.$

The widest layer of the subsets of a set of 3

The divisors of 36 in 3 chains

The maximal elements of the divisors of 24 below 24 itself

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.

the size of the underlying set is a whole number between 2 and 4 ×3
a cyclic order has exactly n arcs of length k ×1
an element's share of the maximal chains is one over its layer's size ×1
and a star is a family of that size ×1
and every chain meets the middle layer exactly once ×1
and every one of them is a whole layer ×1
and every two of its members really do meet ×1
and everything sits below something maximal ×1
and some antichain's shares add to exactly one ×1
and takes one step of size at each rank between its ends ×1
at most k of the n arcs pairwise meet, and k of them do ×1
each chain is symmetric about the middle of the cube ×1
each chain runs upward in the order ×1
each named target is a subset index is a whole number between 0 and 7 ×1
each step adds exactly one element ×1
each subset is joined to the ones one element larger ×1
every chain has an upper bound inside the order ×1
every element of the order is on some chain ×1
every relation of the order is respected by every extension ×1
every subset is on exactly one chain ×1
exactly n + 1 antichains use the chains up exactly ×1
no antichain's shares add to more than one ×1
no chain meets the antichain twice ×1
no element is on two chains ×1
no element is sent to the built set ×1
no subset is on two chains ×1
no two elements of one layer are comparable ×1
so there is something maximal ×1
some incomparable pair splits the extensions between a third and two thirds ×1
the built set disagrees with f(k) about whether k belongs ×1
the chain count is the elements less the matched pairs ×1
the extensions enumerated and the extensions counted by peeling agree ×1
the fewest chains covering the order is the size of the largest antichain ×1
the largest antichain is at least as large as the widest layer ×1
the largest family in which every two sets meet is as large as a star ×1
the map names one subset per element ×1
the number has between four and sixteen divisors ×1
the number has between four and sixteen proper divisors ×1
the number of chains is the size of the middle layer ×1
the number of subsets is two to the power of the set's size ×1
the number whose divisors are ordered is a whole number between 4 and 210 ×1
the number whose divisors are ordered is between 4 and 210 ×1
the number whose proper divisors are ordered is a whole number between 6 and 210 ×1
the order does not already decide everything ×1
the order has an incomparable pair to ask about ×1
the order is one this family draws ×1
the order is one this mode draws ×1
the order is small enough to search every collection of its elements ×1
the order is small enough to search every subset of ×1
the sets of the right size were all generated ×1
the size of the sets in the family is a whole number between 2 and 4 ×1
the theorem needs the sets to be small enough that two can miss each other ×1
the two ends of an edge differ by exactly one element ×1
the view is one the family draws ×1
there are more subsets than elements ×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 list that cannot contain itself

The set of all sets that do not contain themselves is not a set. The argument is the diagonal again, applied to a table whose rows and columns are the same objects, and it destroyed the foundations of mathematics in a postcard.

Analysis

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.

Discrete

Everybody's share of the chains

There are twenty-four ways to build a four-element set one element at a time. Every subset lies on some of them, and no two incomparable subsets share one — so an antichain is a set of disjoint shares of a single whole.

Discrete

How many ways to sort it

An order says some things come before others and leaves the rest open. Counting the orderings consistent with it measures how much is still unknown — and the counting is as hard as any counting problem gets.

Logic

The choice nobody can write down

Given finitely many pairs, picking one thing from each is a finite list of decisions and needs no justification. Given infinitely many, the list cannot be finished — and whether one exists anyway is an axiom, independent of everything else, whose consequences include a theorem most people refuse to believe.

Discrete

The cube cut into chains

Write a subset as a string of brackets, match them the ordinary way, and the unmatched ones say which chain it is on. Six chains cover all sixteen subsets of a four-element set, and the bound and the example arrive together.

Discrete

The largest family that always meets

Change the question from "no two comparable" to "every two share an element" and the answer changes shape. The best antichain is a whole layer; the best intersecting family is a star, and the proof is a circle.

Logic

The row that is not on the list

Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.

Discrete

The widest layer and the longest chain

Order sixteen subsets by inclusion and ask for the largest collection with no two comparable. The answer is the six subsets of size two — the widest layer — and no cleverer collection beats it. Ask instead for the fewest chains covering everything, and the answer is the same number again.

Logic

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.

The whole library · What the figures prove