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

Group Theory & Arithmetic - Lecture 11 (Logic)

Overview

This lecture transitions from the machinery of logic into doing informal proofs inside concrete axiomatic systems. It first reviews what every axiomatic system is built from, then uses group theory as a clean playground for proving theorems from a tiny set of axioms — working through the key group theorems (cancellation, uniqueness of identity, inverses) both in full formal detail and as everyday informal scratch-work. It closes by introducing the axioms of arithmetic and proving foundational facts most people take for granted, such as $1+1 \neq 1$ and $2+2 = 4$ . The payoff is seeing that ordinary algebra is itself just an axiomatic system, and that "obvious" arithmetic truths are genuine theorems with proofs.

Key Concepts

Anatomy of an axiomatic system. Every axiomatic system is built from four kinds of ingredients:

Axioms — the starting assumptions, accepted without proof.
Undefined terms (primitives) — the primary terms used to build everything else; they are not defined but are given meaning by the axioms.
Definitions — abbreviations that bundle primitives together (optional; group theory uses none).
Theorems — the logical consequences of the axioms.

Theorem of logic vs. theorem of a system. A theorem of logic depends on no premises (its premise set is the empty set); it is derivable from the rules of inference alone (rule P, T, CP, US, UG, ES, EG, and identity I) and so may be used inside any system. A theorem of a system (e.g. a group theorem) is entailed by that system's axioms and is only valid where those axioms hold.

The underlying logic. The working logic here is first-order predicate logic with identity. The rule of identity has two parts: everything is identical to itself, and equals may be substituted for equals.

The group axioms. A group is a set $G$ together with an operation $\circ$ satisfying three axioms:

A1 (Associativity): $\forall x,y,z:\ (x \circ y) \circ z = x \circ (y \circ z)$ .
A2 (Identity): $\exists e \in G$ such that $\forall x:\ x \circ e = x$ .
A3 (Inverses): $\forall x\ \exists x^{-1}$ such that $x \circ x^{-1} = e$ .

Here $\circ$ is any operation (not function composition specifically). A running interpretation is the real numbers under addition, where $e$ is $0$ and $x^{-1}$ is $-x$ . The primitives of group theory are exactly the symbols that are not logical symbols: the operation $\circ$ , the constant $e$ , and the inverse symbol $^{-1}$ . (Equality, quantifiers, parentheses, and variables all belong to logic.)

Two reusable theorems of logic. Because identity lets you substitute equals for equals, you may apply the same operation to both sides of an equation:

Theorem A: if $x = y$ then $x \circ z = y \circ z$ (operate on the right).
Theorem B: if $x = y$ then $z \circ x = z \circ y$ (operate on the left).

The nine group theorems (selected results). From A1–A3 one proves, in order:

T1 (Right cancellation): $x \circ z = y \circ z \implies x = y$ .
T2: $x \circ e = e \circ x$ (the identity works on either side).
T3: $x \circ x^{-1} = x^{-1} \circ x$ (an element commutes with its own inverse).
T4 (Uniqueness of identity): $x \circ y = x \implies y = e$ .
T5 (Left cancellation): $z \circ x = z \circ y \implies x = y$ .
T6 (Uniqueness of inverses): $x \circ y = e \implies y = x^{-1}$ .
T7 (Double inverse): $(x^{-1})^{-1} = x$ .
T8 / T9 (Solvability): $\forall x,y\ \exists z$ with $x = y \circ z$ , and likewise $\exists z$ with $x = z \circ y$ .

Formal vs. informal proof. A formal proof lists every rule of inference and tracks which premises each line depends on. An informal proof is the carefully-organized scratch-work — enough reasoning that a reader could reconstruct the formal version — skipping the most tedious mechanical steps. Almost all of mathematics is written informally.

No commutativity. Crucially, group theory has no commutative law: in general $x \circ y \neq y \circ x$ . That is why swapping symbols is the hard part of these proofs, and it is why commutativity is genuinely independent of A1–A3 (function composition is a model where the axioms hold but commutativity fails).

The arithmetic axioms. Arithmetic takes a domain $\mathbb{R}$ with operations $+$ and $\times$ and a relation $<$ . Operators take terms and return terms; a relation takes terms and returns true/false. The fifteen axioms include commutativity and associativity of $+$ and $\times$ , the distributive law $x \times (y + z) = x\times y + x\times z$ , additive identity $0$ and multiplicative identity $1$ , additive inverses (for all $x$ ), multiplicative inverses (for all $x \neq 0$ ), the order axioms for $<$ (asymmetry, transitivity, trichotomy, and compatibility with $+$ and with $\times$ by a positive number), and the non-triviality axiom $0 \neq 1$ . (The one missing axiom is completeness, needed for the irrational numbers and calculus.)

Building the numbers. The only things known to be in $\mathbb{R}$ at first are $0$ and $1$ . The non-triviality axiom guarantees $0 \neq 1$ , which forces the system to be more than a single point. One then proves $1 + 1$ is a new element, names it $2$ , defines $3 := 2+1$ , $4 := 3+1$ , and so on — generating the natural numbers. Asking "what is the definition of $1$ ?" is a category error: $1$ is a primitive, given meaning by the axioms, not defined.

Worked Example

Right-cancellation (Group Theorem 1): prove that $x \circ z = y \circ z \implies x = y$ .

The idea mirrors algebra: to undo $\circ z$ , operate on the right by $z^{-1}$ (Theorem A), regroup with associativity (A1), collapse $z \circ z^{-1} = e$ (A3), and drop the identity (A2).

\begin{aligned} x \circ z &= y \circ z &&\text{(premise)} \\ (x \circ z) \circ z^{-1} &= (y \circ z) \circ z^{-1} &&\text{(Theorem A: operate by } z^{-1}\text{)} \\ x \circ (z \circ z^{-1}) &= y \circ (z \circ z^{-1}) &&\text{(A1: associativity)} \\ x \circ e &= y \circ e &&\text{(A3: } z \circ z^{-1} = e\text{)} \\ x &= y &&\text{(A2: } w \circ e = w\text{)} \end{aligned}

Discharging the premise (rule CP) and generalizing (rule UG) gives the full statement $\forall x,y,z:\ x \circ z = y \circ z \implies x = y$ .

A second taste — proving $2 + 2 = 4$ in arithmetic. Using the definitions $2 = 1+1$ , $3 = 2+1$ , $4 = 3+1$ and associativity of addition:

\begin{aligned} 2 + 2 &= 2 + (1 + 1) &&\text{(definition of } 2\text{)} \\ &= (2 + 1) + 1 &&\text{(associativity)} \\ &= 3 + 1 &&\text{(definition of } 3\text{)} \\ &= 4 &&\text{(definition of } 4\text{)} \end{aligned}

Summary

Every axiomatic system rests on four pillars: axioms, undefined terms (primitives), definitions, and theorems.
A theorem of logic depends on no premises and is usable in every system; a theorem of a system depends on that system's axioms.
A group is a set with one operation satisfying associativity, an identity, and inverses; from just these three axioms flow the cancellation laws, uniqueness of identity and inverses, and the double-inverse and solvability results.
An informal proof is the organized scratch-work behind a formal proof — enough reasoning to reconstruct every step, with the tedious mechanics omitted.
Group operations are not assumed commutative, so swapping the order of terms is never free; commutativity is independent of the group axioms.
Arithmetic is itself an axiomatic system: from its axioms, with $0$ and $1$ as the only given elements, you can build the natural numbers and rigorously prove facts like $1+1 \neq 1$ and $2+2 = 4$ .