Théorème

N1 — Algebraic uniqueness of primes

The primes are the unique atoms of the multiplicative monoid $(\mathbb{N}_{\geq 1}, \times)$.

Statement

The prime numbers $\mathbb{P} = \{2, 3, 5, 7, 11, \ldots\}$ are the unique atoms of the multiplicative monoid $(\mathbb{N}_{\geq 1}, \times)$ . The Eratosthenes sieve is the unique constructive procedure that identifies them by successive elimination of multiples.

Théorème

Plain reading. If you look for the “fundamental bricks” of multiplication on $\mathbb{N}$ , only one family is possible: the primes. Any other candidate reduces to them. And the only constructive way to find them is the sieve. This is what makes PT’s starting point non-arbitrary.

Why it matters

N1 establishes the non-arbitrariness of PT’s starting point. If another family of “bricks” could play the role of the primes, the sieve would be one choice among many. N1 closes that door: no hidden degree of freedom at the root.

It is also N1 that closes the philosophical coherence of the programme: PT does not postulate exotic objects — it uses the bricks arithmetic itself imposes.

Proof — outline

Atoms: by the Fundamental Theorem of Arithmetic, every integer $n \geq 2$ factors uniquely as a product of primes (up to order).
Uniqueness: no other family of multiplicative generators admits unique factorisation on $\mathbb{N}$ .
Constructivity: the sieve eliminates multiples by increasing order; no other constructive procedure with the same axioms gives the same result.

Detailed proof

Part 1 — Primes are atoms

An atom of $(\mathbb{N}_{\geq 1}, \times)$ is an element $a \neq 1$ such that $a = bc \Rightarrow b = 1$ or $c = 1$ . That is exactly the definition of a prime. The Fundamental Theorem of Arithmetic (existence + uniqueness of factorisation) shows that primes multiplicatively generate $\mathbb{N}$ and that no other element has the atomicity property.

Part 2 — Uniqueness of the family

Suppose an alternative family $\mathcal{A} \subset \mathbb{N}$ such that every $n$ factors uniquely as a product of elements of $\mathcal{A}$ . By prime-by-prime factorisation uniqueness, each $a \in \mathcal{A}$ must itself be a product of primes. If $a$ is a non-trivial product (not a single prime), then $a$ has an internal factorisation, contradicting atomicity in $\mathcal{A}$ . So $\mathcal{A} \subseteq \mathbb{P}$ . Conversely, missing a prime $p \in \mathcal{A}$ would make $p$ non-factorisable, contradiction. So $\mathcal{A} = \mathbb{P}$ .

Part 3 — Uniqueness of the sieve

Every constructive procedure identifying primes must decide, for each $n$ , whether $n$ is prime. The minimal method tests divisibility by all primes $p \leq \sqrt{n}$ previously identified. That is exactly the Eratosthenes sieve.

Any variation (elimination in a different order, by other relations) either fails to terminate, produces the same result (if it respects divisibility), or produces an incorrect result. The Eratosthenes sieve is therefore the unique correct and finite constructive procedure.

For the complete derivation, see chapter 2 of the monograph.