What 'never runs out' means
'Infinite' is slippery until you define it without the word 'forever': a set is infinite if every finite list of its members misses something. This flips infinity from a mystical idea into a checkable challenge — hand me any finite list and I produce a member you missed. That is exactly the shape of Euclid's proof about primes.
A hotel where every guest list, no matter how long, turns out to be missing someone standing in the lobby.
The even numbers are infinite: given any finite list of evens, add 2 to the largest one — a new even number not on the list.
Making infinity precise is what lets mathematics prove things like 'primes never run out' rather than merely believe them.
Level 1 The precise statement
A set S is infinite iff for every finite list of elements of S there exists an element of S not in that list.
Level 3 What it stands on (0 direct)
Nothing — this is foundational. It is one of the roots of the graph.
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.
2 downstream results would collapse with it. See the blast radius on the graph →