You can always ask for the next prime
Infinitude plus well-ordering gives an ordering: there is a definite 1st prime (2), 2nd (3), 3rd (5), and an n-th for every n. This turns the primes from a mere set into a sequence you can index, tabulate, and study — the object behind questions like 'how big is the n-th prime?'
Knowing a library has infinitely many books lets you shelve them and give each a call number.
p₁=2, p₂=3, p₃=5, p₄=7, p₅=11 … p₂₅ = 97. The corollary says p₁₀₀₀₀₀₀ exists (it's 15,485,863) before anyone computes it.
Prime-generation code in cryptographic libraries is, at heart, an implementation of this corollary: 'fetch me the next prime after N' is guaranteed to terminate.
Level 1 The precise statement
For every positive integer n there exists an n-th smallest prime.
Level 2 The proof
Level 3 What it stands on (2 direct)
- Well-Ordering Principle (axiom)
- Infinite set (definition)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
- Divisibility respects linear combinations (lemma)
- Every integer > 1 has a prime divisor (lemma)
- Infinitude of primes (Euclid) (theorem)
- The n-th prime exists (corollary)
Level 4 The verified record
This page is generated from a machine-checked node. The kernel confirms its dependencies resolve, nothing is circular, and it grounds in axioms (foundation: ZFC). The content hash below makes tampering evident.
Nothing yet — this is a frontier result.
Nothing depends on it yet, so its failure would be contained.