# 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

Axioms of the Real Numbers - Lecture 12 (Logic)

Quiz

✦ Hall of Aces

Discussion