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Concept

Closure

The smallest set containing a starting collection and shut under a stated set of operations, so applying one never leaves it. It is what makes a definition well posed: without it, applying an allowed operation could take an object outside the set it was defined in.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Post's five classes, and which connectives escape them. A table of connectives against the five closed classes, with the completeness verdict for each.

One connective is enough

Of the sixteen ways to combine two truth values, exactly two can build all the others by themselves. Which two is not obvious, and the reason turns out to be five properties that a connective either has or escapes.

logic · Truth functions Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

computation · Constructible numbers The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

computation · Constructible numbers The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

The field with four elements

The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.

computation · Finite fields Every relabelling of a 4-gon's corners, and the 8 that are motions. All 24 permutations of the corners drawn one by one, with the 8 that preserve every distance marked; the rest deform the polygon and are not symmetries.

Eight ways to leave a square alone

A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not taken on trust here — it is what a search through all twenty-four relabellings of the corners comes back with.

algebra · Symmetry groups Three sets where a fixed point escapes, and one where it cannot. A ring turned about its centre, an open disc halved toward a point of its rim, the plane shifted sideways, and the closed disc turned and shrunk. Only the last has a point that its map leaves where it is.

Where the fixed point escapes

The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.

topology · Fixed points Two hundred and three points from two. After two rounds: 203 points, 11 rational, 76 in ℚ(√3) but not rational, 116 needing a new square root (√2: 20, √5: 22, √7: 14, √11: 36, √13: 20, √35: 4); 10 lines and 16 circles in the second round.

What two points build in two rounds

Start with two points and draw every line and circle they allow; mark every crossing; do it again. The first round gives six points, the second gives 203, and the third gives over 1.7 billion. Computed exactly, the 203 points show the tower of square roots growing in real time — eleven rational points, seventy-six that need √3, and the rest needing one of six new square roots.

computation · Constructible numbers

Named alongside it

The objects these essays reach for when they reach for this one.

Constructible numberField extensionCounterexampleStraightedge and compassAffine functionBasisBoundaryBrouwerCharacteristicCircleCompass and straightedgeConnective

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