You're second person to make this point... maybe it's more confusing than I thought.

You can write statements with letters that define numbers, eg "the smallest number greater than 3" defines 4 (in integers).

Take all 60 letter or shorter statements (finitely many). Toss out those that are gibberish or don't define a integer as they're irrelevant, and you have a (even smaller, though besides the point) finite set of sentences, hence integers. That leaves infinitely many not defined by such statements. There is a smallest positive one. Bingo bongo