The Timpson Lyceum · First-Order Predicate Logic (2024)

Axioms of the Real Numbers - Lecture 12 (Logic)

Axioms of the Real Numbers – Lecture 12

Overview

First-order logic gives us the language for building axiomatic systems — the way we encode systematic bodies of knowledge from a small set of starting assumptions. This lecture develops one such system: arithmetic, treating the real numbers $\mathbb{R}$ as an abstract domain governed by fifteen axioms about two operators ( $+$ , $\times$ ) and one relation ( $<$ ). Alongside the axioms, the lecture introduces informal proof — proofs written in ordinary mathematical style rather than as fully formalized derivations. We see how every familiar arithmetic fact (that $1+1\neq 1$ , that $2+2=4$ , cancellation laws, $x\cdot 0=0$ , the sign rules) must be earned as a theorem from the axioms, with nothing about real numbers assumed in advance.

Key Concepts

Components of an axiomatic system.

Primitives (undefined terms) — the non-logical symbols that appear in the axioms; their meaning comes only from the axioms. Here the primitives are $+$ , $\times$ , $<$ , $0$ , and $1$ . (Compare geometry, where point, line, plane are undefined.)
Axioms — statements assumed true, the starting assumptions.
Theorems — everything that must be true if the axioms are: the logical consequences.
Definitions — new symbols built from primitives. Always eliminable, kept purely for convenience.

Operators vs. relations.

An operator takes arguments from the domain and returns a member of the domain. $+$ and $\times$ are closed binary operators on $\mathbb{R}$ : combining two elements always yields another element of $\mathbb{R}$ . (Contrast subtraction on the naturals: $2-5=-3\notin\mathbb{N}$ , so it is not closed there.)
A relation is a predicate returning true/false. $<$ is a two-place relation on $\mathbb{R}$ .

The 15 axioms (all variables range over $\mathbb{R}$ ):

Commutativity: $x+y=y+x$ and $x\cdot y=y\cdot x$ .
Associativity: $(x+y)+z=x+(y+z)$ and $(x\cdot y)\cdot z=x\cdot(y\cdot z)$ .
Distributivity: $x\cdot(y+z)=x\cdot y+x\cdot z$ .
Additive identity: there exists $0$ with $x+0=x$ for all $x$ .
Multiplicative identity: there exists $1$ with $x\cdot 1=x$ for all $x$ .
Additive inverse: for every $x$ there exists $y$ with $x+y=0$ .
Multiplicative inverse: for every $x\neq 0$ there exists $y$ with $x\cdot y=1$ (zero has none).
Order axioms for $<$ : asymmetry ( $x<y \Rightarrow \neg(y<x)$ ); transitivity ( $x<y \wedge y<z \Rightarrow x<z$ ); trichotomy-of-distinctness ( $x\neq y \Rightarrow x<y \vee y<x$ ); additive compatibility ( $x<y \Rightarrow x+z<y+z$ ); multiplicative compatibility ( $0<z \wedge x<y \Rightarrow x\cdot z<y\cdot z$ ).
Nontriviality: $0\neq 1$ .

Convention on parentheses. Strictly, each operator takes exactly two arguments, so an expression like $x\cdot y+z$ is ambiguous without parentheses. By the agreed convention "multiplication before addition," $x\cdot y+z$ means $(x\cdot y)+z$ . This is a notational agreement, not an axiom.

Equality is logical. $=$ is a symbol of logic (first-order logic with identity), so its rules — reflexivity, substitution of equals for equals, and adding/applying the same operation to both sides ( $a=b \Rightarrow a\circ c=b\circ c$ for any well-defined deterministic operation $\circ$ ) — come for free and need not be re-axiomatized.

Formal vs. informal proof. Formal proofs cite an exact rule at every step and become tedious through repetition. Informal proofs state the key axiom/theorem used at each step in ordinary prose — there are no rigid rules, only style and clarity. Every step must still be justified by an axiom, a prior theorem, or a law of logic.

Building the number system. Starting from only $0$ and $1$ , closure forces new elements into $\mathbb{R}$ : one proves $1+1$ differs from both $0$ and $1$ , defines it as $2$ , then $2+1=:3$ , and so on, giving all the naturals. The additive-inverse definition then yields the integers, and the multiplicative-inverse definition yields the rationals.

Defined symbols.

$-x$ (additive inverse): $-x=y \;:\Leftrightarrow\; x+y=0$ . Well-defined because the additive inverse is unique.
$x-y$ (subtraction): $x-y=z \;:\Leftrightarrow\; x=y+z$ . Provably equal to $x+(-y)$ .
$x^{-1}$ (multiplicative inverse, $x\neq 0$ ): $x^{-1}=y \;:\Leftrightarrow\; x\cdot y=1$ .
$x/y$ (division, $y\neq 0$ ): $x/y=z \;:\Leftrightarrow\; x=y\cdot z$ . Provably equal to $x\cdot y^{-1}$ .

Representative theorems proved.

Uniqueness of identities: if $x+y=x$ then $y=0$ ; if $x\neq 0$ and $x\cdot y=x$ then $y=1$ .
Left cancellation: $x+y=x+z \Rightarrow y=z$ ; and (for $x\neq 0$ ) $x\cdot y=x\cdot z \Rightarrow y=z$ . These are not free from logic — they require an inverse element to exist.
Zero product: $x\cdot 0=0$ .
Uniqueness of inverses (additive and multiplicative), licensing the $-x$ and $x^{-1}$ definitions.
Sign rules: $-(-x)=x$ , $\;(-1)\cdot x=-x$ , $\;(-1)\cdot(-1)=1$ (a lemma), and $(-x)\cdot(-y)=x\cdot y$ .

What's still missing. These axioms give all of basic algebra but not continuity. The one further axiom needed is completeness — every nonempty set bounded above has a least upper bound — which is what makes $\sqrt{2}\in\mathbb{R}$ and underlies all of calculus/real analysis.

Worked Example

Theorem (zero product): for all $x$ , $\;x\cdot 0 = 0$ .

The strategy is to make zero "interact" with multiplication, which only the distributive axiom does. Write $0$ as $0+0$ , distribute, then cancel.

\begin{aligned} x\cdot 0 &= x\cdot(0+0) && \text{(}0+0=0\text{, additive identity)} \\ &= x\cdot 0 + x\cdot 0 && \text{(distributivity)} \\ x\cdot 0 + 0 &= x\cdot 0 + x\cdot 0 && \text{(add identity }0\text{ to the left)} \\ 0 &= x\cdot 0 && \text{(left cancellation of the }x\cdot 0\text{ term)} \end{aligned}

So $x\cdot 0 = 0$ for every $x$ . $\blacksquare$

Theorem (uniqueness of the additive identity): $x+y=x \Rightarrow y=0$ .

Assume $x+y=x$ . By the additive-inverse axiom pick $w$ with $x+w=0$ , then add $w$ to both sides and regroup.

\begin{aligned} w+(x+y) &= w+x && \text{(add }w\text{ to both sides)} \\ (w+x)+y &= w+x && \text{(associativity)} \\ (x+w)+y &= x+w && \text{(commutativity)} \\ 0+y &= 0 && \text{(substitute }x+w=0\text{)} \\ y &= 0 && \text{(additive identity)} \end{aligned}

So $0$ is the only element that behaves like an additive identity. $\blacksquare$

Summary

An axiomatic system has four parts — primitives, axioms, definitions, theorems — and arithmetic is built as one over the domain $\mathbb{R}$ with primitives $+,\times,<,0,1$ .
The fifteen axioms cover commutativity, associativity, distributivity, additive/multiplicative identities and inverses, five order properties for $<$ , and $0\neq 1$ ; everything else about real numbers must be proved.
Informal proofs justify each step by naming the axiom, theorem, or law of logic used, rather than writing a fully formal derivation; equality and "do the same thing to both sides" come from logic, while cancellation must be proved using inverses.
Numbers are forced into $\mathbb{R}$ by closure: the naturals (from repeatedly adding $1$ and proving each result new), the integers (via the unique additive inverse $-x$ ), and the rationals (via the unique multiplicative inverse $x^{-1}$ ).
Key results derived include $x\cdot 0=0$ , uniqueness of identities and inverses, left cancellation for $+$ and $\times$ , and the sign rules $-(-x)=x$ , $(-1)\cdot x=-x$ , and $(-x)\cdot(-y)=x\cdot y$ , with subtraction and division defined in terms of $+$ and $\times$ .
The axioms suffice for ordinary algebra but not for continuity; that requires the additional completeness (least-upper-bound) axiom, the source of everything distinctive about the real numbers.