Axioms: A High School Introduction to Axiomatic Thinking shows how entire branches of mathematics grow from a handful of simple rules. Written for Grade 12 students (and keen Grade 11s), it’s a friendly, rigorous bridge from high-school problem solving to the proof-based thinking you’ll meet in university math!
 
Starting from first principles, you’ll learn the axiomatic method—define your objects, state your axioms, and derive everything else with clear logic. Along the way, you’ll practice clean proof writing and see how classic areas of math fit together.
What you’ll learn
The language of logic: statements, connectives, quantifiers, and common proof styles (direct, contrapositive, contradiction, induction)
Sets and structure: subsets, power sets, relations, functions, and equivalence classes
Numbers by axiom: Peano arithmetic and mathematical induction
Geometry then and now: Euclid’s postulates, non-Euclidean models, and independence
Real numbers: field axioms, order, and completeness (least upper bound)
Algebra by design: groups, rings, and fields from their defining axioms
Probability from three axioms (Kolmogorov) and its core identities
What’s inside
Step-by-step Proof Workshops that model how to start, structure, and polish a proof
Exercises that ramp from quick checks to challenge problems
A capstone project where you build your own “micro-theory” from scratch
Full Solutions Manual for all exercises (student- and teacher-friendly)
Glossary, notation guide, proof templates, and an index for fast review
Who it’s for
Students preparing for proof-based university math (calculus, algebra, analysis)
Teachers seeking a short, rigorous enrichment unit or transition-to-proof resource
Axioms turns curiosity into proof—so you don’t just learn math, you learn how mathematics is built!