Field

Logic — page 3

What can be said in a system, what follows from it, and what it cannot settle about itself.

A countable structure grown by adding witnesses. Stages of a structure built from 0 and 1 by adding sums, products, negatives and roots of quadratics: 2, 4, 12, 158 elements between −3 and 3.

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.

An additive function that is nowhere a line. A square window with 1157 points of the graph of an additive function that sends √2 to 0, scattered across the window and meeting every part of it, beside a dashed diagonal marking the straight line y = x.

A function that adds and is nowhere a line

Every continuous function with f(x + y) = f(x) + f(y) is a straight line through the origin. Drop continuity and, given the axiom of choice, there are others — functions that add perfectly and whose graphs are scattered densely over the whole plane. A finite piece of the construction can be drawn exactly; the whole of it needs a basis of the real numbers that no one can write down.

Infinitely many guessers, finitely many wrong

An infinite line of people each wears a black or white hat, sees every hat in front and none of their own, and must guess their own colour. With a finite line, each guesser is right half the time whatever they agree in advance. With an infinite line and the axiom of choice, they can agree a strategy under which all but finitely many are right — and nobody can carry it out.

Six sentences from two quantifiers

One relation, two variables, 'for every' and 'there is': there are eight ways to arrange them and six different sentences come out. Which of them imply which is a small, complete diagram, found by checking all 512 relations on three points — and the diagram crosses over in the middle, which is where every confusion about the order of quantifiers lives.

"There is an x" is a shadow: x² + ax + 1 = 0 has a solution exactly when |a| ≥ 2. A grid with a horizontal and x vertical, marking the cells the curve x squared plus a x plus one equals zero passes through; beneath it, a strip marking the columns that contain a mark, which are exactly those with a at least two in size.

A quantifier is a shadow

'There is an x such that …' asks whether a column of a grid contains a mark — which is the same as asking whether a shape casts a shadow on the axis below it. Over the real numbers every such shadow can be described without the quantifier, by polynomial inequalities: 'x² + ax + 1 = 0 has a solution' is just a² ≥ 4. Over the whole numbers the same kind of shadow can carve out the primes, and any set a computer can list.

Weighted votes: two functions with a cut, and parity without one. For three functions of three letters, the eight assignments placed on a line by a weighted count, with true assignments filled and the threshold marked where one exists.

A plane through the cube

Some truth functions are weighted votes: give each letter a weight, add the weights of the true letters, and say yes when the total passes a threshold. On the cube of assignments, such a function is a plane cutting the true corners from the false. Majority is one. Exclusive-or is not, and never can be — and of the 65,536 functions of four letters, only 1,882 are. The ones that are are exactly what a single artificial neuron can compute.

9 corners of the 4-cube: some corner always has 2 chosen neighbours. The 4-dimensional cube with 9 of its corners chosen so that no chosen corner has more than 2 chosen neighbours, the fewest possible, with the edges between chosen corners drawn heavy.

Half the cube and √n neighbours

Choose more than half the corners of an n-dimensional cube, any way at all, and some chosen corner has at least √n chosen neighbours. That statement about a cube settled a thirty-year question about how sensitive a truth function must be to its inputs, and its proof is a matrix of plus and minus ones whose square is n times the identity. A search over every choice for the 4-cube finds the bound exactly: nine corners, and some corner always has two chosen neighbours.

A smallest model of ∀x (Ax → ∃y (By ∧ ¬Cy)) ∧ ∃x (Ax ∧ Cx) ∧ ∀x (Bx → ¬Ax). Three overlapping circles with some regions shaded as empty and a single dot in each occupied region, forming a model of a sentence of monadic first-order logic.

One thing in each region is enough

Give first-order logic its full apparatus of nested quantifiers but only one-place predicates, and every question about truth is still settled by the regions of a diagram. A predicate cannot tell apart two things in the same region, so no model ever needs more than one thing per region — and with three predicates there are only 255 models to try.

Carroll's babies and crocodiles, on four ellipses. Four overlapping ellipses labelled with the four classes of a sorites, the regions emptied by its premises shaded, and the regions its conclusion requires to be empty outlined.

The conclusion is what survives the erasing

Lewis Carroll's puzzles give three premises about four classes — babies, logical people, the despised, crocodile-managers — and ask what follows. Draw all four, shade what the premises rule out, then erase the classes the conclusion is not about: a region survives as empty only if everything above it was. What is left is the conclusion, and erasing a class turns out to be exactly one step of resolution.

Adjectives that describe themselves, and the one that cannot exist. A grid of 8 adjectives against the same adjectives as written words, marked where the property holds of the spelling; the diagonal marks self-describing adjectives, and a flipped diagonal row labelled heterological matches no row.

The word that cannot describe itself

Some adjectives describe themselves — 'short' is short — and some do not — 'lengthy' is not lengthy. Call the second kind heterological, and ask whether 'heterological' is heterological. It is exactly when it is not. The diagonal that beat every list of numbers, every set of sets and every provability predicate has been turned on words that describe words, and it proves that no language can contain a word for its own notion of describing, naming or truth.

Whole-number points in a strip, and the shadow they cast. The lattice points satisfying 2x ≤ 5y ≤ 2x + 1 for x from 0 to 30, and their projection onto the x-axis, which repeats every 5.

Arithmetic with addition alone

Over the real numbers, a quantifier's shadow is described by inequalities. Over the whole numbers with addition and multiplication, a shadow can be any set a computer can list. In between lies arithmetic with addition and no multiplication, and there the shadows are always the same kind of thing: a finite exception, then a pattern that repeats. The whole numbers made from coins worth 6, 9 and 20 are every number from 44 on; the squares, which need multiplication, never repeat at all.

Gluing two countermodels: p ∨ ¬p. Two small Kripke models, each refuting one half of a disjunction, and the model made by placing both above a new first stage, at which neither half is forced.

A proof that says which half

Classical logic proves p ∨ ¬p without any idea which half is true. The constructive system never does that: whenever it proves a disjunction, it proves one of the two halves. The reason is a picture — two countermodels placed side by side above a new first stage that forces neither — and the logics between the two lose the property exactly when their pictures are not allowed to be glued.

Which chains of truth values validate Gödel's disjunctions. A grid of Gödel's pigeonhole formulas in m letters against chains of n truth values, marking which chains validate which formula, forming a staircase where m exceeds n.

No table of truth values is enough

Two truth values decide classical logic. Gödel asked in 1932 whether some longer list of values could decide the constructive system, and answered with a pigeonhole: with n values, some two of n + 1 statements must share one, so a formula saying exactly that holds in every n-valued table and is not a theorem. The chains of truth values then descend forever, and what they share is a logic of its own — the logic of the real interval.

Tseitin's clauses on the cube. the cube with a variable on each edge and a charge of 0 or 1 at each vertex, exactly one vertex charged 1. The parity demands give 32 clauses that cannot all be true.

A contradiction that is only a sum

Put a variable on every edge of a graph and ask each vertex for an odd or an even number of true edges, with the demands adding up to odd. Add all the demands and every edge is counted twice, so the left side is zero and the right side is one: the contradiction is a single sum. Resolution cannot add. It has to reach the same conclusion clause by clause, and on a graph where every group of vertices has many edges leaving it, that takes exponentially long.

Unifying f(x, g(x)) with f(h(y), g(z)). Three term trees: f(x, g(x)), f(h(y), g(z)), and their common instance f(h(y), g(h(y))) under the most general unifier x ↦ h(y), z ↦ h(y).

Two terms made equal, and no more

Resolution with variables needs two literals to clash, and they clash only after something has been substituted for their variables. There are infinitely many substitutions that would do. One of them is the most general — every other is it followed by something more — and an algorithm of four rewriting rules finds it or proves there is none. That single computation turns the search for instances from guessing into arithmetic.

A program whose output is its own text. The 32-character program (f=>f(f))(f=>"(f=>f(f))("+f+")") beside its output, which is identical.

The program that prints itself

The diagonal argument has always been used to destroy — to show that a list misses something, that a sentence cannot be proved. Run the same move the other way and it builds. Kleene's recursion theorem says every program can be given its own text to work with, and the proof is a program that prints itself, thirty-two characters long, which can be run and checked.

No algorithm can read what a program does

Whether a program halts cannot be decided by any algorithm. Henry Rice showed in 1953 that the halting problem is not special: no algorithm can decide any property of what a program does — whether it ever prints a 7, whether it computes the successor function, whether it is a virus — except the two properties that hold of every program or of none. Every one of the 20,736 smallest Turing machines can be checked by hand; the theorem says why that stops.

$The rationals and the dyadic fractions, matched 12 times without a crossing. Two number lines from 0 to 1, rationals above and dyadic fractions below, with 12 back-and-forth matchings: 1/2↔1/2, 1/3↔1/4, 2/3↔3/4, 1/4↔1/8, 3/4↔7/8, 2/5↔3/8, 1/5↔1/16, 3/5↔5/8, 4/5↔15/16, 2/7↔3/16, 1/6↔1/32, 3/8↔5/16.$

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.

The first 16 vertices of Rado's graph, joined by binary digits. Adjacency table and circular drawing of the Rado graph on vertices 0 to 15, where i < j are adjacent when bit i of j is 1; 32 edges.

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.

$Three sizes of infinity, and the collections that have each. ℵ₀: the fractions, the algebraic numbers, every finite text, every definition of a number; 2^ℵ₀: the real numbers, the continuous functions, the open sets, the closed sets, the Borel sets; 2^(2^ℵ₀): all functions, all sets of reals, the non-Borel sets, the measurable sets.$

What counting can prove exists

There are only as many continuous functions as points of a line, only as many Borel sets, only countably many definitions — and more sets of reals than any of those. So counting proves that discontinuous functions, non-Borel sets and undefinable numbers exist without exhibiting one. It cannot prove that a set with no length exists, because the sets that have a length are exactly as numerous as all sets, and that difference turns out to be the difference between what the axioms force and what they merely allow.

A countable closed set stripped of its isolated points, 3 times. Cantor–Bendixson derivatives of the set of sums of up to 2 reciprocals: sizes 750, 40, 1, 0 in the truncation, ending empty.

Closed sets obey the continuum hypothesis

Whether some set of reals is bigger than the whole numbers and smaller than the line is a question the axioms cannot answer. For closed sets it has an answer, found in 1883: strip away the isolated points, again and again, and what is left is either nothing or a perfect set, which is as large as the line — so every closed set is countable or of the line's size. The same answer holds for Borel and analytic sets, and fails to be provable exactly one step further up.

Addition as a machine with two states. Two-state carry automaton over columns (x, y, z): from carry 0 staying on 000 011 101, to carry 1 on 110; from carry 1 staying on 010 100 111, back on 001.

A machine that carries one bit

Write three numbers in base two, one above another, and read the columns from the right. Whether the bottom number is the sum of the top two can be checked with one bit of memory — the carry. That two-state machine is the whole of addition, and machines can be combined, negated and made to guess a missing number. So every sentence about whole numbers built from addition can be decided by building a machine for it, and the same machines can also handle 'x is a power of two', which addition alone cannot say.

Three-colourability as a sentence: there exist three sets such that …. Petersen graph coloured with three colours; witness sets R = {0,2,6}, G = {1,3,5,9}, B = {4,7,8}; all 25 first-order checks pass.

There is a relation such that

A graph can be coloured with three colours exactly when there are three sets of its points such that every point lies in one and no edge joins two points of the same set. The sets are a guess; once guessed, the rest is a list of simple checks. Ronald Fagin proved in 1974 that this shape of sentence — 'there exist relations such that' followed by first-order conditions — describes exactly the problems whose solutions can be checked quickly, the class called NP. The central question of computer science is therefore also a question about what sentences can say.

Smallest circuits for four functions of three letters. Four small circuit diagrams, each with three inputs feeding a few two-input gates, computing parity, a selector, majority and exactly-one-true with two, three, four and four gates.

How many gates a truth table needs

Every truth function can be built from gates, and the natural measure of a function is the fewest gates that build it. For three letters the whole answer can be computed: no function needs more than four. For n letters, a count shows that almost every function needs about 2ⁿ/n gates, and a construction shows that none needs more. And for any function anyone can actually write down, the best proof in fifty years says it needs 3.1n.

All essays