Axioms, definitions and theorems

Why this exists

A proof gives a reason for a statement, and the reason is an earlier statement. Ask “why?” of that one, and you are handed an even earlier one. The questions cannot run backwards for ever, and they cannot go round in a circle, because a circle proves nothing. Somewhere the answers have to stop.

The statements where they stop are called axioms. Choosing them openly is what lets mathematics say exactly what it is talking about. Euclid built the geometry of the plane on a short list of starting statements, and every theorem in it can be traced back to that list. In the nineteenth century people changed one item on the list and found a second, different geometry that works just as well as the first. Around 1900 a harsher lesson arrived in the theory of sets: a rule that sounded perfectly obvious, that any description picks out a set, turned out to contradict itself. Both stories say the same thing. Whatever you start from has to be stated and examined, because you cannot see what a proof depends on unless you can see where it starts.

The words in a proof need the same care. Two people argue about whether 0 is even. One says it cannot be, because even numbers are the ones you can share out in pairs, and there is nothing to share. The other says it must be, because it sits between -1 and 1, and both of those are odd. They can go on all evening. Nothing either of them says can settle it, because each is using the word “even” to mean a slightly different thing. The cure is to agree, in words that leave no room, on exactly what the word means. After that the argument is over, because it is no longer an argument about words. You hold the agreed meaning up against the number and look. That is what a definition is for. It turns a word that has a feel into a rule that has a verdict. “Rectangle”, “prime”, “continuous” and “derivative” are each, before anything is proved about them, one exact sentence, and every proof about them begins from that sentence. A proof has no way of knowing what a word “really” means. It knows only what the word was defined to mean, and that is all it needs.

What comes out of axioms and definitions are proved statements, and books label them: Theorem 2.3, Lemma 2.2, Corollary 2.4. All three kinds have proofs, and all three are true for the same kind of reason, so the labels are not telling the reader how sure to be. They say what a statement is for. A proof of anything large is hard to read as one long wall of reasoning, so it is cut into pieces: smaller facts proved first, then the main result, then the easy things that fall out of it. Each piece has a different job, and the three words name the job. Without them a long text is a pile of true statements in a row. With them, the reader can see what leads to what.

Definition

A mathematical system is built from three kinds of statement. Axioms are accepted without proof. Definitions fix what new words mean. Theorems are proved from the axioms and the definitions, and among proved statements the words lemma and corollary name particular jobs. This section takes the three in turn.

What an axiom is

An axiom is a statement that a mathematical system accepts without proof and uses as a starting point. Three things make a statement an axiom:

• it is stated before any proving begins,
• any proof in the system is allowed to use it as a reason,
• the system does not prove it. The system chooses it.

The older word postulate names the same thing, and many geometry books still use it. An axiom usually talks about objects that the system leaves undefined, such as “point” and “line” (see undefined terms). The axioms are then the only thing the system knows about those objects, and everything else has to come from them.

A statement that is proved from the axioms is a theorem. The difference between the two is only where they sit: a theorem has reasons beneath it, and an axiom is where the reasons end.

Here is a system small enough to hold in your head. It has three axioms, about two kinds of thing, points and lines.

Axiom 1. There are exactly three points.

Axiom 2. Any two different points lie on exactly one line.

Axiom 3. Every line has exactly two points on it.

Nothing else is known. A line here is not straight or long, and not even drawn. It is whatever the axioms say it is. Figure 1 is one picture of a world that obeys all three.

What a definition does

A definition introduces a word by stating, in words already understood, the exact condition something must meet to be called by it.

Even. An integer n is even if n = 2k for some integer k.

The sentence has three parts: the word being defined (even), the kind of thing it applies to (an integer n), and the condition (there is an integer k with n = 2k). Geometry’s words work the same way.

Rectangle. A rectangle is a quadrilateral with four right angles.

Here the word is rectangle, the kind of thing is quadrilateral, and the condition is four right angles.

Three features are worth knowing before you use one.

It uses only words already understood. In the definition of even, the words “integer” and “times” were settled earlier. Every word inside a definition has to have been defined before, or else be one of the few terms a subject accepts without definition, such as the “point” and “line” of the axiom system above.

“If” means “if and only if”. The definition is written “is even if n = 2k”, but it means more than that: an integer is even when n = 2k for some integer k, and only then. Definitions are usually written with a bare “if” and always read as a two-way statement. Why that reading is forced, and what it has to do with Biconditional statements, is taken up in converses and biconditional definitions.

It is not true or false. A definition does not report a fact that might turn out wrong. It is a decision about how a word will be used. There is nothing to prove about it. You can accept it, or you can choose a different one, and nobody has made a mistake either way.

That is how a definition differs from an axiom, though neither is proved. An axiom makes a claim about its objects, so a given world can keep it or break it, as the one-line world of Example 6 breaks Axiom 3. A definition makes no claim at all. It only gives a name to a condition, so no world can break it.

Theorem, lemma and corollary

There are three words for proved statements, and each names a job.

A theorem is a proved statement that a text presents as one of its main results: worth stating, remembering and citing for its own sake.

A lemma is a proved statement that is there chiefly to help prove something else. It is a stepping stone.

A corollary is a proved statement that follows quickly from a result just proved, often a special case of it or a different way of saying it. Its proof is short, because the work has already been done: cite the result and add a line or two.

Many books also use proposition for a proved statement that carries less weight than a theorem. Where the line falls is the author’s choice. Older usage differs again: the usual English editions of Euclid’s Elements call every numbered result a proposition, great or small, as in Book I, Proposition 15.

The first part of this section used the word theorem for any statement proved from the axioms. That is the wide meaning, and it still holds: a lemma and a corollary are theorems in the wide sense, because each is proved. The word in the list above is the narrow meaning, a label for the main results. When this lesson writes “theorem” beside “lemma” and “corollary”, it means the narrow one.

Here is one of each, all about integers. They are proved in Examples 16 to 18.

Lemma. Every square of an integer leaves remainder 0 or 1 when divided by 4.

Theorem. A sum of two squares of integers never leaves remainder 3 when divided by 4.

Corollary. The equation x^2 + y^2 = 2027 has no solution in integers.

The lemma is there because the theorem needs it. The theorem is the result worth having. The corollary is one instance of the theorem: 2027 leaves remainder 3 when divided by 4, so it is one of the numbers the theorem rules out.

Intuition

Axioms as the rules of a game

Think of the axioms as the rules of a game. A rule of chess is not true or false; it is what makes the game that game. Once the rules are fixed, a statement counts as true in the system exactly when it follows from them, and nothing else counts. Someone else may play a game with a different rule, and that is not a mistake by either player. It is a different game.

Figure 2 shows what “follows from” looks like. Each theorem stands on earlier statements. Follow any arrow down and you reach an axiom, and no arrow leads into an axiom, because an axiom is not supported by anything in the system.

There is a second picture worth keeping beside this one. A set of axioms describes a kind of world: any collection of things, with whatever the axioms mention, in which every axiom comes out true. Figure 1 is a world for the three axioms above. One set of axioms can have many worlds: Axioms 2 and 3 on their own hold in the three-point world and also in a world of four points (Example 2). The axioms of Euclid’s plane hold in the plane built out of pairs of real numbers, which a flat sheet of paper pictures well, but a different set of axioms is needed for the surface of a ball. A drawing of a world is only a picture of it. What the world contains is settled by the axioms, not by how the drawing looks.

A definition as a membership test

A definition is a membership test, written out in full. The word names everyone who passes the test and nobody else.

Here is the test for even, as a picture. Stand at 0 on a number line and hop by 2. After k hops to the right you land on 2k, and after k hops to the left you land on -2k. Saying “n = 2k for some integer k” says that n is one of the landing points.

The landing points are \dots, -4, -2, 0, 2, 4, 6, \dots. The number 7 is not one of them: three hops reach 6 and four reach 8, so 7 is jumped over. And 0 is one, because it is where the hopping starts: zero hops, k = 0, and 0 = 2 \cdot 0. The old argument about zero is settled without any argument, by running the test.

The test also works in both directions. If you are told a number is a landing point, you may write it as 2k. If you can write it as 2k, you may say it is a landing point. That two-way freedom is what makes a definition so useful inside a proof.

Theorems, lemmas and corollaries as jobs

Figure 4 is built like Figure 2, but each proved statement’s box now names its job. It shows a small text arranged the way the three words suggest. Read it from the bottom.

Two lemmas are proved first, each straight from the axioms and definitions. The theorem is then proved from the two lemmas: its proof is short because the lemmas did the heavy lifting. Two corollaries are proved last, each in a line or two from the theorem. An arrow always points from a reason to what it supports, and every chain of arrows ends at the axioms.

Lemma and corollary describe a statement’s place beside another statement. We say “Lemma 1 is a lemma for Theorem 3”, and “Corollary 4 is a corollary of Theorem 3”. So the same statement can be both at once. In Example 14 a result about quadrilaterals is a corollary of the triangle angle sum theorem, and then in Example 15 it is used as a lemma for a result about pentagons.

From the writer’s side, here is when a step is worth pulling out as a lemma.

• It is long, and giving it a name and its own proof keeps the main argument short enough to follow.
• It is needed more than once. Proving it a single time beats repeating the same work in every place it is used.
• It is interesting on its own. Then it may deserve to be remembered, and the name follows the importance.

A corollary is the reverse move. After a big result, the immediate consequences are written down once, in plain view, so the reader sees what the result buys without having to rediscover it.

The labels are also instructions for reading. A theorem is what to take away. A lemma can be skipped on a first pass, and trusted until you want to know why it is true. A corollary is worth a moment to check that it really is a quick step from its parent.

Results

Worked examples

Examples 1 to 6 work inside small axiom systems. Examples 7 to 12 test, unfold and fold definitions. Examples 13 to 19 build theorems, lemmas and corollaries and show each doing its job.

Boundaries

What an axiom is

An axiom is accepted, not announced as true of the real world. The axioms of the plane say what must be so if something is to count as a Euclidean plane. Whether a chalkboard, or a soccer field, or space itself follows them is a separate question, answered by measuring and not by proving.

A set of axioms must not contradict itself. A system that proves a statement and also its opposite is useless, since in ordinary logic it then proves every statement whatever. Showing that a set of axioms is free of contradiction usually means exhibiting a world in which all of them hold, which is what Figure 1 does for the three-point axioms.

Accepted without proof is not accepted without reason. Axioms are chosen because they are simple, few, and strong enough to build what is wanted. But they can be replaced. Swap one, and you are describing a different world, not correcting a mistake.

The axioms do not always settle every question. Example 5 found a statement that Axioms 2 and 3 leave open. The same thing happens at a far larger scale. A result of Gödel, stated here without proof, says this: take any system whose axioms do not contradict each other, whose axioms can be listed by a fixed rule (a computer program could print them all), and which can prove the basic facts about adding and multiplying whole numbers. Then there is a statement about whole numbers that the system can neither prove nor refute. (Gödel proved it in 1931 with a slightly stronger assumption in place of “do not contradict each other”; Rosser removed that extra assumption in 1936.)

An axiom is not a conjecture. A conjecture is a statement nobody has proved yet, and it is hoped that a proof exists. An axiom is a statement nobody intends to prove.

What a definition does

A definition says which things a word names. It does not say there are any. The sentence “call an integer huge if no integer is larger than it” is a perfectly good definition, with a clear test. But for any integer n, the integer n + 1 is larger, so nothing passes the test. Whenever it matters that a defined thing exists, that has to be shown separately (see Existence proof).

A definition says only what its condition says. The definition of rectangle asks for four right angles. That the diagonals of a rectangle are equal is true, but it is not in the definition, so it cannot be used until it has been proved. The same goes the other way: a feeling about what a word suggests, such as “a rectangle is longer than it is wide”, is not part of it.

A definition covers only the kind of thing it names. Even is defined for integers. The number 2.5 is not an integer, so the definition does not make it even, and the usual convention is that only integers are called even or odd.

It cannot refer to itself. Suppose a word were defined by “an integer is bonky if it is one more than a bonky integer”. To test whether 5 is bonky you would ask whether 4 is, then 3, and so on without end. The test never reaches a place where the question is answered. Every word in a definition has to be settled before the definition begins, and at the very bottom some terms are left undefined on purpose, exactly as the points and lines of an axiom system are.

Two definitions of one word may both be in use. When each condition implies the other, they name exactly the same things and either may be chosen, the other becoming a theorem. When they do not, as with the two trapezoids of Example 11, the word means different things in different books, and nothing can be carried between them unchecked.

Theorem, lemma and corollary

The words describe a role. They do not measure difficulty or importance. A lemma may be much harder than the theorem it serves, and a name can stay after the role has changed. Zorn’s lemma is still called a lemma, yet in standard set theory it is known to be equivalent to the axiom of choice, which makes it a cornerstone rather than a small helper.

The labels are a judgment, not a test. The word even has an exact condition that any integer either meets or fails. “Main result”, “helper” and “quick consequence” have no such condition, so a result can be a theorem in one book, where it is the point of a chapter, and a lemma in another, where it is a step toward something bigger. How big a step a corollary may be is a judgment of the same kind, as Examples 13 and 14 show. What is not a matter of judgment is whether the statement is proved.

Names can outlive their meaning. The statement called Fermat’s Last Theorem was named that long before anyone had a proof of it, and it was proved only in the 1990s. So a name, taken alone, does not tell you whether a proof exists. Inside a text, a statement labeled a theorem is a promise that a proof follows.

Only proved statements get these labels. A conjecture, an axiom and a definition, all met above, have no proof, so none of them is a theorem, a lemma or a corollary.

A corollary reaches only as far as its proof. It follows from its parent, perhaps with facts already proved, and can claim nothing those do not support. In particular the converse of a theorem is not a corollary of it: it is a separate statement that needs its own proof, and it may be false, as Example 18 shows.

Where this goes next

Once you know where a system starts and what its words mean, you can read how a text is built. What may be assumed and what must be shown looks at what a proof may start from, and the answer always includes the axioms and the definitions of its words. A direct proof is unfold, reason, fold, chained from the starting statements to a conclusion, and what to do when that will not work is proof by contradiction, which Example 3 used. A long proof is usually a theorem standing on lemmas, each proved by one of those methods.

Proving statements about even and odd numbers and Proving statements about divisibility are the definition moves applied to two particular definitions, and the tidy example is what they look like with the name changed. The counting view of even numbers is even and odd numbers, and factors and multiples holds the definition that most of number theory starts from.

For geometry in particular, converses and biconditional definitions sorts every statement in a proof into what was agreed, what was named, and what was shown, and explains why a definition reads both ways. The axioms of geometry start from the words left undefined, which the lesson on undefined terms describes, and the most famous of them, the one that Example 5 echoes, is the parallel postulate. Two places where the exact definition matters are trapezoids and rectangles. The cut in Examples 14 and 15 is the method of Angle sums of polygons, and the lemma style of the last example continues in degree and the handshake lemma.

Exercises

Reading an axiom system

Use the three axioms of the three-point world for Exercises 1 to 4.

1. How many lines pass through the point P?
2. How many points do two different lines of this world have in common?
3. Is “some line contains all three points” provable, refutable or undecided? Explain.
4. Is “the point P is to the left of the point Q” provable, refutable or undecided by these axioms? Explain.

Counting in other worlds

For Exercises 5 to 9 keep Axioms 2 and 3 but change the number of points.

5. A world has exactly 4 points. How many lines are there, and how many lines pass through each point?
6. A world has exactly 5 points. How many lines are there, and how many lines pass through each point?
7. A world has exactly 12 points. How many lines are there?
8. A world has exactly 45 lines. How many points does it have?
9. Can a world of this kind have exactly 20 lines? Explain.

Axioms for numbers

Use R1 to R3 from Example 4, together with R4: 1 \cdot a = a for every number a.

10. Use R2 to compute 7 \cdot 13 + 7 \cdot 87 without multiplying 7 by 13.
11. Write 97 as 100 + (-3) and use R2 to compute 6 \cdot 97.
12. Show that a + (-1) \cdot a = 0 for every number a.

Worlds and independence

13. A world has four points A, B, C, D, and its lines are the four sets \{A, B, C\}, \{A, B, D\}, \{A, C, D\} and \{B, C, D\}. How many lines hold both A and B? Which of Axioms 2 and 3 fail in this world?
14. A world has three points P, Q, R and exactly two lines, \{P, Q\} and \{P, R\}. Which of Axioms 1, 2 and 3 hold? What does this show about Axiom 2?
15. On a globe, let a point be a place on the surface and a line be a great circle, such as the equator or the circle through both poles along the 0^\circ and 180^\circ meridians. Does every pair of points lie on exactly one line? Are there two lines that never meet?

Testing against a definition

16. Use the definition of even to decide whether each integer is even: 26, -14, 91, -1.
17. An integer is a multiple of 5 if it equals 5k for some integer k. Decide whether each is a multiple of 5: 35, -20, 0, 52.
18. An integer is odd if it equals 2k + 1 for some integer k. Decide whether each is odd: 15, -3, 0, 100.
19. A positive integer is a perfect square if it equals k^2 for some positive integer k. Decide whether each is a perfect square under this definition: 49, 50, 1, 0.

Geometry words

20. A rhombus is a quadrilateral whose four sides all have the same length. Use this definition to decide whether a square is a rhombus, and whether every rhombus is a square. Give an example that settles the second question.
21. A quadrilateral has three right angles. The four angles of any quadrilateral add to 360^\circ. Find the fourth angle, and say whether the shape is a rectangle under the definition in Example 8.
22. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Using the definition of trapezoid with “at least one pair of parallel sides”, decide whether every parallelogram is a trapezoid, and whether every trapezoid is a parallelogram. For the second question, settle it with a quadrilateral whose corners you give as coordinates.

Unfolding and folding

23. Let n be an even integer. Show that n + 6 is even.
24. Show that for every integer n, the number 10n - 8 is even.
25. Show that for every integer n, the number 4n^2 + 2n is even.
26. Let m and n be even integers. Show that m - n is even.
27. Let n be an even integer. Show that 3n is even.
28. Let m be an even integer and n any integer. Show that mn is even.

Words made up for the occasion

For Exercises 29 to 32, an integer is a zorp if it equals 5k + 2 for some integer k.

29. Decide whether each is a zorp: 17, -3, 25, 2.
30. Show that 5n + 7 is a zorp for every integer n.
31. Show that the sum of a zorp and a multiple of 5 is a zorp.
32. Show that the product of two zorps equals 5k + 4 for some integer k, and that no product of two zorps is a zorp.

Corollaries of the angle sum

33. One acute angle of a right triangle is 28^\circ. Find the other.
34. The angles of a triangle are x^\circ, (x + 10)^\circ and (x + 20)^\circ. Find all three.
35. Three angles of a convex quadrilateral are 88^\circ, 112^\circ and 67^\circ. Find the fourth.
36. The four angles of a convex quadrilateral are x^\circ, 2x^\circ, 3x^\circ and 4x^\circ. Find all four.
37. Four angles of a convex pentagon are 110^\circ, 120^\circ, 100^\circ and 130^\circ. Find the fifth.
38. Cut a triangle off a convex pentagon in the way of Example 15 to find the sum of the angles of a convex hexagon, and then each angle of a regular hexagon.
39. A triangle has angles of 50^\circ at A and 65^\circ at B. Side BC is extended past C, and the angle between the extension and side CA is called the exterior angle at C. Find it.

Squares and remainders

40. What remainder does 37^2 leave when divided by 4?
41. The theorem of Example 17 rules out some of the numbers 2027, 2029, 2031 and 2035 as sums of two squares. Which ones? Then write the remaining number as a sum of two squares.
42. What remainders can the cube n^3 of an integer leave when divided by 4? Write n = 4m + r with r one of 0, 1, 2, 3.
43. Show that the square of an integer leaves remainder 0 or 1 when divided by 3. Write n = 3m + r with r one of 0, 1, 2.
44. Use Exercise 43 to prove: if x and y are integers and x^2 + y^2 is divisible by 3, then both x and y are divisible by 3.
45. Use Exercise 44 twice to find every pair of non-negative integers x \le y with x^2 + y^2 = 2025.

Build a lemma, then a theorem

46. Show that the square of an integer leaves remainder 0, 1 or 4 when divided by 8. Use n = 4m + r with r one of 0, 1, 2, 3.
47. Use Exercise 46 to show that a sum of three squares of integers never leaves remainder 7 when divided by 8. Then write down two numbers that cannot be written as a sum of three squares.
48. The theorem of Example 17 rules out remainder 3 when dividing by 4. Give a number that leaves remainder 1 when divided by 4 and is not a sum of two squares.

The handshake

49. Explain why no graph has five vertices with degrees 3, 3, 3, 2 and 2.
50. The degrees 3, 3, 3, 1 pass the test of the corollary: there are four odd degrees. Show that even so, no graph has four vertices with these degrees.

Reasoning about axioms

51. Explain why a proof cannot consist entirely of statements that each have a proof inside the same proof.
52. A friend says an axiom is “a statement that is obviously true”. Give two reasons that this is not what makes a statement an axiom.
53. Suppose the axioms of a system prove a statement and also its opposite. What does this say about the system?
54. A statement is proved from Axioms 1 and 2. A student then finds a world where both axioms hold and the statement is false. What must be true?
55. One axiom of a system can be proved from the others. Is something wrong?

Reasoning about definitions

56. A friend says zero cannot be even “because there is nothing to pair up”. Answer the friend using the definition, and say why the feeling about pairs does not settle the matter.
57. Someone asks you to “prove that the definition of even is correct”. Explain why that cannot be done, and name two things it does make sense to ask about a definition.
58. A student writes: “Let m and n be even. Then m = 2k and n = 2k, so m + n = 4k = 2(2k), which is even.” What has this argument actually covered, and how is it repaired?
59. Write a circular definition of a made-up word, and explain what goes wrong when you try to decide whether a particular number meets it.
60. Call an integer huge if no integer is larger than it. Is that a legitimate definition? Is any integer huge?
61. One book defines a rectangle as a parallelogram with four right angles. Another defines it as a quadrilateral with four right angles. Show that both books name exactly the same shapes. You may use this fact: if two lines are cut by a third line, and the two interior angles on one side add to 180^\circ, the two lines are parallel.

Reasoning about theorems, lemmas and corollaries

62. An author proves that every square leaves remainder 0 or 1 when divided by 4, uses that in the proof of one later result, and never mentions it again. Which of the three words suits it best, and why?
63. A friend says that “a lemma is a theorem that is not quite as true”. Explain what is wrong with that, and what the three words do distinguish.
64. A lemma is written to help prove a theorem. Explain why the proof of the lemma cannot use that theorem.
65. A result is called a lemma in one book and a theorem in another. Can both books be right?
66. In Example 14 the diagonal AC was assumed to lie inside the quadrilateral. A dart-shaped quadrilateral has one angle greater than 180^\circ, at a corner C, and its diagonal BD lies outside it. Which diagonal can you use to repeat the argument, and why does the argument then still give 360^\circ?
67. Prove from the angle sum theorem that an exterior angle of a triangle equals the sum of the two interior angles at the other two corners.