Every number is 0, or one step past another number. From that single move — S, "the next one" — and two recursive rules, addition and multiplication are built, not assumed. Here you pick a and b and watch the successors stack: a+b and a·b unfold from Peano's axioms alone, live. Terms go in, the recursion runs, the answer comes out. The blue team builds it; the red team tries to make 2+3 lie.
source Peano, Arithmetices principia, nova methodo exposita (1889) — archive.org/details/arithmeticespri00peangoog. Rendered, not quoted.
A number is a structure, not a symbol. 0 is the empty box {s:null}; S(n) wraps n in one more box. So 3 is literally S(S(S(0))).
Addition and multiplication are then defined, each by a base case and a step:
add(a,0)=a · add(a,S b)=S(add(a,b))
mul(a,0)=0 · mul(a,S b)=add(mul(a,b),a)
Every other sphere that counts, sums, weighs or indexes takes ℕ for granted. This is the sphere that hands it to them: 0 and S, plus induction, generate all of it.
Boole's algebra of 0 and 1 and Aristotle's 256 forms both assume a settled domain of objects — this sphere is where the objects come from. Each sphere is the next one's premise; the successor is theirs.
The blue team's live check: rebuild every add(a,b) and mul(a,b) from the successors and confirm each equals ordinary arithmetic, for all a,b in 0..8. If red tampers, this badge is where it shows.
The inputs are two naturals a, b ∈ 0..8 — nothing more. Each is fed in as a tower of successors over a single 0:
| you pick | the machine sees | = |
|---|---|---|
| 0 | 0 | zero |
| 1 | S(0) | one step |
| 2 | S(S(0)) | two steps |
| n | Sn(0) | n steps |
No number is stored as a digit; each is a chain you can count. That chain is what the engine below rewrites — and the depth of the answer's chain is the answer.
a =
b =
Change a, b, or the operation — the answer is recomputed by unfolding the successor axioms every time, never looked up in a table.
What the machine produces, proven: for every pair a,b ∈ 0..8 — 81 pairs — both add and mul reduce through the successor axioms alone to exactly ordinary a+b and a·b. 162 equalities, checked live, plus: S is injective and 0 is no one's successor.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
And by Gödel's second theorem (1931), PA cannot prove its own consistency. This engine's 0..8 window is honest for a deeper reason too: unary successors cost one box per unit — S64(0) for 8·8 — so the representation is exponentially large. It is correct, not practical; the world's arithmetic runs on positional digits this sphere silently justifies.
"Peano invented these axioms." Cut. Dedekind stated essentially the same characterization a year earlier (Was sind und was sollen die Zahlen?, 1888); Peano cites him. What is Peano's is the notation.
"Peano started the numbers at 0." Cut. The 1889 original begins at 1. Starting at 0 is the later, now-standard convention — the one this engine uses; the axioms work from either.
"The recursive add and mul are Peano's." Kept, corrected. The step-definitions of + and × come from Grassmann (1861); Peano set them inside a formal axiom system. The recursion here is Grassmann's, made runnable.
The red team's move: rewrite the addition step so it wraps two successors instead of one — add(a,S b)=S(S(add(a,b))). Every sum then overcounts, and multiplication (built on add) rots with it. The blue team's witness (window 7) is watching.