Generator

A line, a point, and many parallels

A generator in the logic library, called 31 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.

disc-model 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

A line, a point, and many parallels. A disc whose lines are arcs meeting the boundary at right angles, showing several lines through one point that never meet a given line.

A countable structure grown by adding witnesses

Real numbers that are roots of quadratics with coefficients up to 5

Sentences the real line and the algebraic reals agree on

A bounded set of algebraic numbers with no least upper bound among them

The same two sets, equal in size outside a model and unequal inside it

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 1 axioms c > 0, …, c > 0 are satisfied by 1 and not by 0 ×6
(a + b) + c = a + (b + c) holds on every sampled triple ×1
(a × b) × c = a × (b × c) holds on every sampled triple ×1
−1 appears at the first stage ×1
√2 appears at some stage drawn ×1
8 to 32 vertices drawn ×1
a + b = b + a holds on every sampled triple ×1
a × (b + c) = a × b + a × c holds on every sampled triple ×1
a × b = b × a holds on every sampled triple ×1
a < b and 0 < c imply a × c < b × c holds on every sampled triple ×1
a < b implies a + 1 ≤ b holds on every sampled triple ×1
a < b implies a + c < b + c holds on every sampled triple ×1
a ≠ 0 implies a = d + 1 for some d holds on every sampled triple ×1
a line drawn as parallel was checked and does not meet the base line inside the disc ×1
a triangle has three vertices ×1
and by the last every request is answered ×1
and never reaches the upper half ×1
and to no vertex of V ×1
between 4 and 9 terms ×1
between three and eight pieces, on a line long enough to hold them ×1
between two and four levels of galaxies ×1
each crossing point was computed and lies on both lines ×1
each dot is a root of its polynomial ×1
each lower truncation is below π and each upper one above ×1
each neighbourhood shows 2 to 4 steps either side ×1
each stage adds new elements ×1
every angle of the triangle is positive ×1
every drawn polynomial belongs to the structure ×1
every function from A to B is listed ×1
every geodesic arc meets the boundary circle at a right angle ×1
every ordinary number is one or the other ×1
every partial matching preserves order both ways ×1
every vertex is inside the disc ×1
exactly one of a < b, a = b, b < a holds on every sampled triple ×1
exactly one of the lines through the point never meets the base line ×1
halving preserves order ×1
heights from 2 to 7 ×1
inside the model no function is onto ×1
more than one geodesic through the point misses the base line ×1
more vertices answer more requests ×1
no drawn dot is a known transcendental ×1
nothing lies strictly between an element and its successor ×1
outside, the bijections number size factorial ×1
sets of two to four elements ×1
some element is neither even nor odd ×1
the angles of a triangle in this model add to less than a straight angle ×1
the chosen vertices are joined exactly as Petersen's are ×1
the defect is a positive area ×1
the drawn elements are in increasing order ×1
the enumeration of the other side reaches every gap ×1
the even witness doubles back to the element ×1
the galaxies are distinct and in order ×1
the graph has a vertex joined to exactly the right earlier ones ×1
the lower ones rise and the upper ones fall ×1
the matching keeps every edge and every non-edge ×1
the model's functions stay not-onto after relabelling B ×1
the number of lines tried through the point is a whole number between 3 and 9 ×1
the number of parallels drawn is a whole number between 1 and 9 ×1
the odd witness gives the element ×1
the outside point is inside the disc ×1
the parity statement fails on sampled elements ×1
the point is off the base geodesic ×1
the random graph has a vertex of the needed kind ×1
the rule is symmetric ×1
the view is one of triangle, euclid, compactness, polymodel, parity, axioms, galaxies, algebraic, skolemhull, relative, sentences, supremum, backforth, bfsteps, forthonly, bfalgebraic, endpoints, gap, rado, radowitness, radorandom, radoembed, radoiso, radoresilient ×1
the witness for "∀a∀b∀c ∃x x³ + ax² + bx + c = 0" checks ×1
the witness for "∀x (0 < x → ∃y y·y = x)" checks ×1
the witness for "∀x∀y (x < y → ∃z (x < z ∧ z < y))" checks ×1
the witness for "∃x (x·x = x + x ∧ ¬ x = 0 ∧ ¬ x = 1 + 1)" checks ×1
the witness for "∃x x·x = −1" checks ×1
the witness for "∃x x·x = 2" checks ×1
the witness for "∃x x·x·x = x + 1" checks ×1
the witness is joined to every vertex of U ×1
the witness is not among the deleted vertices ×1
three to nine elements of the structure ×1
two or three stages ×1
X and its whole neighbourhood exceed every ordinary 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 countable field that passes for the line

The real numbers are uncountable, and every first-order sentence about their addition, multiplication and order is also true of a countable field inside them — the real algebraic numbers. Löwenheim and Skolem showed this is no quirk of the reals: every theory with an infinite model has a countable one, including set theory, which then contains sets it calls uncountable.

Logic

A number larger than every number

Ask for a number bigger than 0, bigger than 1, bigger than 2, and so on for ever. Every finite piece of that request is granted by an ordinary number, so compactness grants all of it at once — in a structure that satisfies every sentence true of the whole numbers and still contains something beyond all of them. Nothing in first-order logic can say 'and nothing else'.

Logic

The graph that coin tosses always make

Take infinitely many vertices and toss a coin for every pair to decide whether they are joined. The result is random in every detail — and, with probability one, it is always the same graph. The same graph can be written down without any coins, by joining two numbers when one binary digit of the larger is a one.

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.

Geometry

The triangle that a globe gets wrong

On a sphere, a right triangle with legs of fifty and sixty degrees has a hypotenuse of seventy-two, not seventy-eight. The theorem is not approximately true there — it is false, and what replaces it says exactly how much room the surface has.

Logic

Two lists that are one order

The rationals and the fractions with a power of two below are different sets of numbers, and as orders they are exactly the same — any two countable orders that are dense and have no ends can be matched, point for point, keeping every comparison. The proof is a zigzag, and it settles every question the language of order can ask.

Logic

Two worlds that both obey the rules

A statement is independent of a list of axioms when there is a structure satisfying the axioms where it holds and another where it fails. That is not a claim about what nobody has managed to prove — it is a proof that nobody can.

The whole library · What the figures prove