Axioms and Postulates

Definitions

• An axiom (or postulate) is a statement accepted without proof as a starting point for deduction.
• A theorem is a statement deduced from the axioms by the rules of logic.
• A lemma is a theorem proved chiefly as a stepping stone toward a larger theorem, rather than for its own interest.
• An axiom system is a list of axioms together with the undefined terms they speak about.

Undefined Terms Come With the Axioms

Definitions have the same regress problem as proofs: every definition uses earlier words. So each axiom system also leaves a few terms undefined, such as point and line in geometry, set and is an element of in set theory, and successor in arithmetic. The axioms are the only thing we know about them. As Euclidean Geometry puts it, an undefined term is constrained by the axioms that mention it, never described.

The Classical Distinction

Aristotle separated the common axioms, shared by every science, from the principles proper to one science. Euclid's Elements (around 300 BCE) follows the same line. It opens with five Common Notions, truths about equality and magnitude that hold in any subject, and five Postulates that belong to geometry alone. Proclus discussed the distinction at length in his commentary on Euclid in the fifth century.

Modern Usage Merged the Words

Today the two words are near-synonyms, and which one a system gets is mostly historical habit. Nobody hesitates to say "the group axioms", although they concern groups alone. The same statement about parallel lines is called both the parallel postulate and Playfair's axiom, and Peano's list is called both the Peano axioms and the Peano postulates.

This page keeps the classical meanings, because they sort the material cleanly: Part A is the axioms and Part B the postulates. But no mathematical fact depends on the choice of word.

The Distinction That Matters Today

The division that does real work in modern mathematics is between two different jobs an axiom system can do:

• Foundational axioms describe one intended universe and assert that certain things are true of it. ZFC describes the universe of sets; Peano's postulates describe the natural numbers. A foundational axiom is a claim.
• Structural axioms are a definition in disguise. "A group is a set with an operation satisfying G1 to G4" asserts nothing is true. It picks out a kind of structure, and there are many structures of that kind.

A quick test: does the system describe the objects, or define a kind of object? Commutativity, \(ab = ba\), is an axiom for abelian groups and simply false for the symmetries of a triangle, and nobody concludes the triangle is wrong. It just is not abelian.

The Five Common Notions

1. CN1 Things which are equal to the same thing are also equal to one another. If \(a = c\) and \(b = c\), then \(a = b\).
2. CN2 If equals be added to equals, the wholes are equal. If \(a = b\) and \(c = d\), then \(a + c = b + d\).
3. CN3 If equals be subtracted from equals, the remainders are equal. If \(a = b\) and \(c = d\), then \(a - c = b - d\).
4. CN4 Things which coincide with one another are equal to one another. Figures that can be moved to fit exactly on top of each other are equal. This is the principle behind Euclid's "superposition" proofs.
5. CN5 The whole is greater than the part. A proper part of something is strictly smaller than the whole.

What Became of Them

• CN1 to CN3 became part of the logic of equality. The next section proves CN1 from two axioms about \(=\).
• CN4 was the weak point. Moving a figure is not a notion the other axioms explain, so Hilbert replaced superposition with explicit congruence axioms (Section 8). The site's SAS Congruence page takes the other modern route: it assumes rigid motions and proves SAS from them.
• CN5 survives for finite sets, where it is a theorem, and fails for infinite ones.

The Traditional Laws of Thought

• Identity: \(P \to P\). A statement implies itself.
• Non-contradiction: \(\neg(P \wedge \neg P)\). No statement is both true and false.
• Excluded middle: \(P \vee \neg P\). Every statement is true or false.

These were long treated as the axioms of logic. In a modern formal system they are theorems, derived from a smaller and less obvious set of axioms.

A Standard Axiom System for Propositional Logic

Every formula of one of these three shapes is an axiom, whatever formulas \(P\), \(Q\), \(R\) stand for. Each line is an axiom scheme: a pattern with infinitely many instances.

1. L1
   \[P \to (Q \to P)\]
   A true statement is implied by anything.
2. L2
   \[(P \to (Q \to R)) \to ((P \to Q) \to (P \to R))\]
   Implication distributes over implication.
3. L3
   \[(\neg Q \to \neg P) \to (P \to Q)\]
   A contrapositive proves the original statement.

There is one rule of inference, modus ponens: from \(P\) and \(P \to Q\), infer \(Q\). For quantifiers, two more axiom schemes and one more rule are added:

• \(\forall x\, \varphi(x) \to \varphi(t)\): what holds for everything holds for any particular term \(t\);
• \(\forall x\,(\psi \to \varphi(x)) \to (\psi \to \forall x\, \varphi(x))\), when \(x\) does not occur free in \(\psi\);
• generalisation: from a proof of \(\varphi(x)\) with \(x\) arbitrary, infer \(\forall x\, \varphi(x)\).

Equality

Equality has just two axioms:

1. E1 \(\forall x\; (x = x)\) Reflexivity: everything equals itself.
2. E2 \(x = y \to (\varphi(x) \to \varphi(y))\), for every formula \(\varphi\) Substitution: equals can replace equals in any statement.

The System Is Exactly Strong Enough

Two properties make this system the right one. It is sound: every theorem is true in every interpretation. And by Gödel's completeness theorem (1929) it is complete: every formula true in every interpretation is a theorem. So "provable" and "true in every structure" coincide. Do not confuse this with the incompleteness theorems of Section 12, which are about what particular axiom systems, such as arithmetic, can prove. The practical rules of inference built on this system are in Logic and Quantifiers.

Why Not Just "Every Property Defines a Set"?

That principle, called naive comprehension, is inconsistent. Applied to the property \(x \notin x\) it produces Russell's set \(R = \{x : x \notin x\}\), with \(R \in R \iff R \notin R\). See Sets, Relations and Functions and Russell's Paradox. ZFC replaces the single naive principle with ten careful ones.

The Ten Axioms

1. Z1 Extensionality.
   \[\forall A\, \forall B\, [\forall x\,(x \in A \leftrightarrow x \in B) \to A = B]\]
   A set is determined by its elements, so \(\{1, 2\} = \{2, 1, 1\}\).
2. Z2 Empty Set.
   \[\exists E\, \forall x\, (x \notin E)\]
   There is a set with no elements. By Z1 there is only one, written \(\varnothing\).
3. Z3 Pairing.
   \[\forall a\, \forall b\, \exists P\, \forall x\, [x \in P \leftrightarrow (x = a \vee x = b)]\]
   For any \(a, b\) the set \(\{a, b\}\) exists. Taking \(a = b\) gives \(\{a\}\), and the ordered pair is defined as \((a, b) = \{\{a\}, \{a, b\}\}\).
4. Z4 Union.
   \[\forall F\, \exists U\, \forall x\, [x \in U \leftrightarrow \exists A\,(A \in F \wedge x \in A)]\]
   The elements of the elements of \(F\) form a set, \(\bigcup F\). In particular \(A \cup B = \bigcup \{A, B\}\).
5. Z5 Power Set.
   \[\forall A\, \exists P\, \forall x\, [x \in P \leftrightarrow x \subseteq A]\]
   All subsets of \(A\) form a set, \(\mathcal{P}(A)\). By Cantor's theorem it is always strictly larger than \(A\).
6. Z6 Separation (a scheme: one axiom for each formula \(\varphi\)).
   \[\forall A\, \exists B\, \forall x\, [x \in B \leftrightarrow (x \in A \wedge \varphi(x))]\]
   From an existing set \(A\) you may carve out \(\{x \in A : \varphi(x)\}\). This gives intersections and differences. It is the safe replacement for naive comprehension.
7. Z7 Replacement (a scheme). If for each \(x \in A\) there is exactly one \(y\) with \(\varphi(x, y)\), then those \(y\) form a set:
   \[\forall A\, [\forall x \in A\; \exists!\, y\; \varphi(x, y) \to \exists B\, \forall x \in A\; \exists y \in B\; \varphi(x, y)]\]
   The image of a set under any definable rule is a set: \(\{F(x) : x \in A\}\).
8. Z8 Infinity.
   \[\exists I\, [\varnothing \in I \wedge \forall x\,(x \in I \to x \cup \{x\} \in I)]\]
   There is an infinite set. It is the only axiom that says so, and without it every set could be finite.
9. Z9 Foundation.
   \[\forall A\, [A \neq \varnothing \to \exists x \in A\; (x \cap A = \varnothing)]\]
   Every non-empty set has an \(\in\)-minimal element. Consequences: no set is an element of itself, and there is no infinite descending chain \(\cdots \in x_2 \in x_1 \in x_0\).
10. Z10 Choice. For every set \(F\) of non-empty sets there is a function \(c : F \to \bigcup F\) with \(c(A) \in A\) for every \(A \in F\). From any collection of non-empty sets you can choose one element from each, all at once, even with no rule for choosing.

ZF is Z1 to Z9; ZFC adds Choice.

About the Axiom of Choice

Choice is the one axiom that asserts something exists without describing it. Over ZF it is equivalent to Zorn's Lemma and to the Well-Ordering Theorem (every set can be well-ordered). It is needed to show that every vector space, infinite-dimensional ones included, has a basis. It also has strange consequences, the best known being the Banach–Tarski paradox: a solid ball can be cut into finitely many pieces and reassembled into two balls of the same size.

Gödel (1938) showed that Choice cannot be disproved from ZF, and Cohen (1963) that it cannot be proved from ZF. It is independent, like the Continuum Hypothesis, and mathematicians adopt it because it is useful.

The Five Postulates

1. P1 \(1 \in \mathbb{N}\) There is a first natural number.
2. P2 \(\forall n \in \mathbb{N}:\; S(n) \in \mathbb{N}\) Every natural number has a successor.
3. P3 \(\forall n \in \mathbb{N}:\; S(n) \neq 1\) 1 is not the successor of anything: counting never loops back to the start.
4. P4 \(\forall m, n \in \mathbb{N}:\; S(m) = S(n) \to m = n\) Different numbers have different successors: two counts never merge.
5. P5 Induction. If \(K \subseteq \mathbb{N}\), \(1 \in K\), and \(n \in K \to S(n) \in K\) for every \(n\), then \(K = \mathbb{N}\). Nothing is in \(\mathbb{N}\) except what counting up from 1 reaches.

What Each Impostor Shows

• Stops (fails P2): without P2, counting could end.
• Loops to 1 (fails P3): a four-hour clock satisfies P1, P2, P4 and P5. P3 is what stops arithmetic being clock arithmetic.
• Merges (fails P4): \(2\) and \(5\) share the successor \(3\), so the count joins a loop part-way along.
• Stray chain (fails P5): P1 to P4 all hold, since nothing loops or merges, yet there are extra elements that counting from 1 never reaches. Only induction rules them out.

Theorem: No Finite Set Satisfies P2, P3 and P4

By P2 and P4, \(S\) is a one-to-one map from \(\mathbb{N}\) to itself, and by P3 its image misses \(1\). So \(S\) is a bijection from \(\mathbb{N}\) onto a proper subset of itself. For a finite set that is impossible: a one-to-one map from a finite set into itself is onto. \(\blacksquare\)

So any model of Peano's postulates is infinite in exactly Dedekind's sense from Section 3, and in set theory it is the Axiom of Infinity that makes a model exist.

Arithmetic From the Postulates

Addition and multiplication are defined by recursion on the successor:

Each rule says how to handle \(1\) and how to step from \(b\) to \(S(b)\). Induction (P5) is exactly what guarantees this defines \(a + b\) and \(a \cdot b\) for every \(b\).

Going Further: First-Order Arithmetic

P5 quantifies over all subsets \(K\) of \(\mathbb{N}\). With P5 in that form, Dedekind proved that any two structures satisfying P1 to P5 are isomorphic: the postulates pin down \(\mathbb{N}\) completely. Peano Arithmetic, the version used in logic, replaces P5 by an induction scheme, one axiom for each formula. That version has non-standard models containing elements larger than every ordinary number. They come from the compactness theorem, a corollary of Gödel's completeness theorem, and no first-order list of axioms can avoid them.

The Field Postulates

The Order Postulates and Completeness

1. O1 For all \(a, b\), exactly one of \(a \lt b\), \(a = b\), \(b \lt a\) holds. Trichotomy.
2. O2 If \(a \lt b\) and \(b \lt c\), then \(a \lt c\). Transitivity.
3. O3 If \(a \lt b\), then \(a + c \lt b + c\). Adding preserves order.
4. O4 If \(a \lt b\) and \(0 \lt c\), then \(ac \lt bc\). Multiplying by a positive number preserves order.
5. C Completeness. Every non-empty subset of \(\mathbb{R}\) that is bounded above has a least upper bound. The least upper bound of \(S\), its supremum \(\sup S\), is an upper bound of \(S\) that is below every other upper bound.

Completeness Is What Separates \(\mathbb{R}\) From \(\mathbb{Q}\)

\(\mathbb{Q}\) satisfies every field and order postulate. It fails only C. Take \(A = \{q \in \mathbb{Q} : q^2 \lt 2\}\), which is non-empty (\(1 \in A\)) and bounded above (by \(2\)). It has no least upper bound in \(\mathbb{Q}\):

• Let \(u\) be a rational upper bound. Then \(u \geq 1 \gt 0\), and \(u^2 \neq 2\) because \(\sqrt{2}\) is irrational.
• Put \(v = \dfrac{2u + 2}{u + 2}\), again rational. A direct computation gives \[ v - u = \frac{2 - u^2}{u + 2}, \qquad v^2 - 2 = \frac{2(u^2 - 2)}{(u + 2)^2}. \]
• If \(u^2 \lt 2\), then \(u \in A\), and \(v \gt u\) with \(v^2 \lt 2\), so \(v \in A\) and \(u\) was not an upper bound after all.
• If \(u^2 \gt 2\), then \(v \lt u\) with \(v^2 \gt 2\), so \(v\) is a smaller upper bound, and \(u\) was not the least.

Either way no rational number is the least upper bound. In \(\mathbb{R}\), Completeness supplies one, \(\sqrt{2}\).

Uniqueness, and Equivalent Forms

Any two complete ordered fields are isomorphic, so the postulates describe a single structure. That makes them foundational in the sense of Section 2. Inside ZFC, \(\mathbb{R}\) is built from \(\mathbb{Q}\) by Dedekind cuts, and the postulates become theorems.

Given the field and order postulates, C is equivalent to the Monotone Convergence Theorem, and to the Bolzano–Weierstrass Theorem. It is also equivalent to "every Cauchy sequence converges" together with the Archimedean property. See Monotone Convergence and Bolzano–Weierstrass and Cauchy Completeness.

Euclid's Five Postulates

In Heath's translation, Euclid asks that we may assume:

1. E1 To draw a straight line from any point to any point.
2. E2 To produce a finite straight line continuously in a straight line.
3. E3 To describe a circle with any centre and distance.
4. E4 That all right angles are equal to one another.
5. E5 That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclidean Geometry states these in modern form. Its first postulate adds that the line is unique, which Euclid used without stating, and its fifth is Playfair's equivalent version below.

Playfair's Axiom, and Other Equivalents

Playfair's axiom: through a point not on a given line there is at most one line parallel to it. The first four postulates already prove that at least one parallel exists: construct equal alternate angles, and the weak exterior angle theorem, which needs no parallel postulate, shows the two lines cannot meet (see Euclidean Geometry). So Playfair's axiom is often stated as "exactly one".

Given the first four postulates, each of the following is equivalent to the fifth:

• Playfair's axiom;
• the angles of a triangle sum to \(180^\circ\) (see Triangle Angle Sum);
• there exist two similar triangles that are not congruent;
• the Pythagorean theorem.

The Gaps in Euclid

Euclid's first proof constructs an equilateral triangle from two circles and assumes the circles meet, which nothing in his list guarantees. His proof of SAS moves one triangle onto another, a motion the postulates never mention. And he relies throughout on diagrams for which point lies between which, a notion he never axiomatised. David Hilbert's Foundations of Geometry (1899) filled every gap.

Hilbert's Twenty Axioms

Undefined terms: point, line, plane. Undefined relations: lies on, between, congruent (\(\cong\)). "Two points" always means two distinct points.

I. Incidence (8)

1. I1 For any two points there is a line containing both.
2. I2 For any two points there is at most one line containing both.
3. I3 Every line contains at least two points, and there are three points not all on one line.
4. I4 Any three points not on one line lie in some plane, and every plane contains at least one point.
5. I5 Any three points not on one line lie in at most one plane.
6. I6 If two points of a line lie in a plane, every point of the line lies in that plane.
7. I7 If two planes have a point in common, they have at least one more point in common.
8. I8 There are four points not all in one plane.

II. Order (4)

1. II1 If \(B\) is between \(A\) and \(C\), then \(A\), \(B\), \(C\) are three distinct points of one line, and \(B\) is also between \(C\) and \(A\).
2. II2 For any two points \(A\) and \(C\), there is a point \(B\) on line \(AC\) such that \(C\) is between \(A\) and \(B\). Lines go on: every segment can be extended.
3. II3 Of any three points on a line, at most one is between the other two.
4. II4 Pasch's axiom. Let \(A\), \(B\), \(C\) be three points not on one line, and \(a\) a line in their plane passing through none of them. If \(a\) passes through a point of segment \(AB\), it also passes through a point of segment \(AC\) or of segment \(BC\). A line that enters a triangle must leave it.

III. Congruence (5)

1. III1 Given a segment \(AB\), a point \(C\) on a line \(m\), and a side of \(C\) on \(m\), there is a point \(D\) on that side with \(AB \cong CD\). Any length can be laid off along any line.
2. III2 If \(CD \cong AB\) and \(EF \cong AB\), then \(CD \cong EF\).
3. III3 Suppose \(B\) lies between \(A\) and \(C\), and \(E\) between \(D\) and \(F\). If \(AB \cong DE\) and \(BC \cong EF\), then \(AC \cong DF\). Congruent pieces add up to congruent wholes.
4. III4 Given an angle, a ray from a point \(O\) in a plane, and a side of that ray's line, there is exactly one second ray from \(O\) on that side making an angle congruent to the given one. Every angle is congruent to itself. Any angle can be laid off, in exactly one way.
5. III5 If triangles \(ABC\) and \(DEF\) have \(AB \cong DE\), \(AC \cong DF\) and \(\angle BAC \cong \angle EDF\), then \(\angle ABC \cong \angle DEF\). A weak form of SAS. Full SAS, with the third side and the third angle, is then a theorem.

IV. Parallels (1)

1. IV Given a line \(a\) and a point \(A\) not on it, there is at most one line in their plane through \(A\) that does not meet \(a\). Playfair's form. Existence follows from the other axioms, so only uniqueness is assumed.

V. Continuity (2)

1. V1 Archimedes. For any segments \(AB\) and \(CD\) there is a natural number \(n\) such that \(n\) copies of \(CD\), laid end to end from \(A\) along the ray through \(B\), reach beyond \(B\).
2. V2 Line completeness. The points of a line cannot be enlarged by further points in a way that keeps its order and congruence relations and keeps axioms I to III and V1 true. A line has no gaps. This is the geometric twin of the Completeness postulate for \(\mathbb{R}\), and it is what guarantees Euclid's two circles meet.

The list changed across editions. E. H. Moore showed in 1902 that one of the original order axioms followed from the others, and the completeness axiom was added after the first edition. The twenty above are the standard final form.

School Postulates: Birkhoff and SMSG

G. D. Birkhoff (1932) took the real numbers as given and needed only four postulates: the ruler postulate (the points of a line correspond one-to-one with the reals, and distance is the difference of coordinates); the point–line postulate (two points lie on exactly one line); the protractor postulate (rays from a point correspond to angle measures); and a similarity postulate (SAS for similarity).

The School Mathematics Study Group adapted this in the 1960s into about two dozen postulates, including the ruler, protractor and SAS postulates. That is where most students first meet the word "postulate". The SAS Congruence page proves SAS from rigid motions instead, and the SSS and ASA pages build on it.

Non-Euclidean Geometry

Around 1830 Bolyai and Lobachevsky, and privately Gauss, developed hyperbolic geometry. It keeps every axiom except the parallel axiom, which it replaces by "through a point not on a line there are at least two lines not meeting it".

Elliptic geometry (Riemann, 1854) goes further: there are no parallels at all. It has to change more than the parallel postulate, because on a sphere two antipodal points lie on infinitely many great circles and lines have finite length. The usual fix is to treat each pair of antipodal points as a single point.

Group

A set \(G\) with an operation \(\cdot\) such that:

1. G1 Closure: \(a \cdot b \in G\) for all \(a, b \in G\);
2. G2 Associativity: \((a \cdot b) \cdot c = a \cdot (b \cdot c)\);
3. G3 Identity: there is \(e \in G\) with \(e \cdot a = a \cdot e = a\) for all \(a\);
4. G4 Inverses: each \(a\) has \(a^{-1} \in G\) with \(a \cdot a^{-1} = a^{-1} \cdot a = e\).

The group is abelian if also \(a \cdot b = b \cdot a\). Full treatment in Groups.

Ring and Field

A ring is a set with operations \(+\) and \(\cdot\) such that \((R, +)\) is an abelian group, multiplication is associative with an identity \(1\), and multiplication distributes over addition on both sides. It is commutative if \(ab = ba\). This follows the convention of Rings and Fields; some authors do not require a \(1\).

A field is a commutative ring with \(1 \neq 0\) in which every non-zero element has a multiplicative inverse. That is exactly the field postulates A1 to D of Section 7. Here they define a class that includes \(\mathbb{Q}\), \(\mathbb{C}\) and \(\mathbb{Z}/p\), not one particular field.

Vector Space

A set \(V\) with vector addition and multiplication by scalars from a field \(F\), such that for all \(\mathbf{u}, \mathbf{v}, \mathbf{w} \in V\) and \(a, b \in F\):

These are the eight axioms of Vector Spaces.

Inner Product

On a real vector space, an inner product \(\langle \mathbf{u}, \mathbf{v} \rangle\) is a real number satisfying:

• symmetry: \(\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle\);
• linearity: \(\langle a\mathbf{u} + b\mathbf{w}, \mathbf{v} \rangle = a\langle \mathbf{u}, \mathbf{v} \rangle + b\langle \mathbf{w}, \mathbf{v} \rangle\);
• positive definiteness: \(\langle \mathbf{v}, \mathbf{v} \rangle \geq 0\), with equality only when \(\mathbf{v} = \mathbf{0}\).

Length and angle are then defined from it, and the Cauchy–Schwarz inequality is a theorem of these three axioms alone.

Determinant

The determinant is the function of the rows of a square matrix that is D1 multilinear in the rows, D2 alternating (zero when two rows are equal), and D3 normalised (\(\det I = 1\)). These axioms are unusual: they define a single function, because exactly one function satisfies them. See Determinants and Existence and Uniqueness of the Determinant.

The Setting and the Axioms

A sample space \(\Omega\); a collection \(\mathcal{F}\) of subsets called events, which contains \(\Omega\) and is closed under complements and countable unions; and a function \(P : \mathcal{F} \to \mathbb{R}\) such that:

1. K1 \(P(E) \geq 0\) for every event \(E\). Non-negativity.
2. K2 \(P(\Omega) = 1\). Something certainly happens.
3. K3 If \(E_1, E_2, E_3, \ldots\) are pairwise disjoint, then \(P\left(\bigcup_{i=1}^{\infty} E_i\right) = \sum_{i=1}^{\infty} P(E_i)\). Countable additivity.

Probability works on finite sample spaces and states K3 for two disjoint events. On a finite space that is all K3 ever says, and the derivation below shows it follows from the countable form.

What the Axioms Leave Open

The axioms say how probabilities must behave, not what they are. Long-run frequency, degree of belief and physical propensity all satisfy K1 to K3, so the same theorems serve every interpretation. That neutrality is part of why Kolmogorov's axioms were adopted so widely.

Equivalence Relations and Orders

A relation \(\sim\) on a set is an equivalence relation if it is reflexive (\(a \sim a\)), symmetric (\(a \sim b \to b \sim a\)) and transitive (\(a \sim b\) and \(b \sim c\) give \(a \sim c\)). Every equivalence relation splits its set into disjoint classes; see Sets, Relations and Functions and Equivalence Relations and Partitions.

A relation \(\preceq\) is a partial order if it is reflexive, antisymmetric (\(a \preceq b\) and \(b \preceq a\) give \(a = b\)) and transitive. It is a total order if in addition any two elements are comparable.

• \(\leq\) on \(\mathbb{R}\) is a total order.
• \(\subseteq\) on the subsets of a set is a partial order but not total: \(\{1\}\) and \(\{2\}\) are incomparable.
• Divisibility on \(\mathbb{N}\) is a partial order: \(2\) and \(3\) are incomparable.

Metric Spaces

A metric on a set \(X\) is a function \(d : X \times X \to \mathbb{R}\) such that for all \(x, y, z\):

1. M1 \(d(x, y) \geq 0\), with \(d(x, y) = 0\) exactly when \(x = y\);
2. M2 \(d(x, y) = d(y, x)\);
3. M3 \(d(x, z) \leq d(x, y) + d(y, z)\) The triangle inequality: a detour is never shorter. See the triangle inequality for real numbers.

Topological Spaces

A topology on a set \(X\) is a collection \(\mathcal{T}\) of subsets, called open sets, such that:

1. T1 \(\varnothing\) and \(X\) are open;
2. T2 any union of open sets is open;
3. T3 any finite intersection of open sets is open.

Every metric gives a topology: the open sets are the unions of open balls. The word "finite" in T3 matters. The open intervals \((-1/n, 1/n)\) intersect in \(\{0\}\), which is not open in \(\mathbb{R}\). Topology is the most economical axiom system on this page: three axioms about one kind of set are enough to define continuity.

Models

A model of an axiom system is a structure in which every axiom is true. \(\mathbb{Z}_4\) under addition is a model of the group axioms; the Euclidean plane is a model of Hilbert's axioms; the four structures in Figure 4 are models of four Peano postulates each. Three questions can be asked of any axiom system, and models answer all three.

Consistency

A system is consistent if no contradiction can be derived from it. An inconsistent system proves every statement, so it is worthless. Naive comprehension was inconsistent (Section 5).

A model proves consistency, since a true structure contains no contradictions, but only relative to the setting the model is built in. Beltrami (1868), and later Klein and Poincaré, built models of hyperbolic geometry inside Euclidean geometry. That shows hyperbolic geometry is consistent if Euclidean geometry is.

Independence

An axiom is independent of the others if neither it nor its negation can be proved from them. To prove it, exhibit two models of the others: one in which it holds and one in which it fails.

• The parallel postulate, by the Euclidean plane and the Poincaré disk (Figure 11).
• Each Peano postulate, by the impostors of Figure 4.
• The Axiom of Choice from ZF: Gödel (1938) and Cohen (1963).
• The Continuum Hypothesis from ZFC, by the same two.

Completeness of a Theory

A system is complete if for every statement in its language, either the statement or its negation is provable. Structural systems are incomplete on purpose. The group axioms cannot decide \(ab = ba\), since \(\mathbb{Z}_4\) satisfies it and the symmetries of a triangle do not. Try both in the axiom checker.

For foundational systems, completeness was the hope, and Gödel's incompleteness theorems (1931) ended it. Any consistent axiom system that can be listed mechanically and includes basic arithmetic has statements it can neither prove nor refute. And such a system cannot prove its own consistency. This applies to Peano Arithmetic and to ZFC.