Matpic
mathematics, in pictures
Generator

relation-grid

A generator in the logic library, called 18 times across 5 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.

relation-grid 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

j is one more than i, counting round — as a grid, with both quantifier readingsA grid of marks for a relation, with the row and column facts the two quantifier orders ask about.ji123456123456∀i ∃j : trueevery row carries at least one mark∃j ∀i : falsesome one column is marked all the way downthe relation "j is one more than i, counting round", on 6 rows and 6 columnsevery row has a mark: yes · some column is all marks: no

show: "membership"

Does this set contain that one — and the row that is missingA membership table with the diagonal marked, and beneath it the complement of the diagonal, which is not among the rows.S1S2S3S4S5S6S7S1S2S3S4S5S6S7Rthe sets that do not contain themselvesrow i, column j is marked when set i contains set j — the diagonal asks whether a set contains itselfthe row beneath is the complement of the diagonal, and it is not one of the rows above it

show: "diagonal"

The diagonal, and the row built to be off the listA table of rows of ones and zeros with the diagonal marked, and beneath it the row obtained by flipping every diagonal entry.S1011010100000S2110101000001S3001111100001S4101010000010S5000100100011S6011111000011S7111001100100S801010000010010011001neweach row is a set of whole numbers, as its membership row; the marked squares are thediagonalthe row underneath is the set containing n exactly when the nth set does not — so no listof sets contains every set of whole numbers

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.

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 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.

 Logic

An infinite tree has an infinite path

A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

 Logic

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

 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.

 Logic

The sentence that says it has no proof

Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.

The whole library · What the figures prove