Peano Arithmetic
A standard formal axiom system for ordinary arithmetic — natural numbers, +, ×, and induction
Peano Arithmetic (abbreviated PA) is a formal system — a fixed set of axioms and rules of inference — for reasoning about the natural numbers (0, 1, 2, …), addition, multiplication, and mathematical induction.
In this series, PA is a concrete example, not a prerequisite. When we say a formal system is “rich enough for arithmetic,” we mean something in the same league as PA: strong enough to express ordinary counting and induction, so Gödel-style incompleteness applies — there are arithmetical truths such a system cannot prove if it is consistent.
How PA fits the epistemic ladder:
- Tier 0 (Rⁿ): the mathematical structure itself — not the same as any formal system.
- Tier 1: axiom statements and proof steps formalised or committed in a log (adoption of PA for an organisation; a checked proof object).
- Tier 2+: theorems relative to those tier-one records — e.g. “provable in PA,” not “identical to the Real.”
PA is not set theory. For foundations that encode most of classical mathematics in one system, see ZFC.