Peano builds ℕ · pairs of naturals give ℤ · pairs of integers give ℚ · Dedekind cuts give ℝ · runs locally
classify a number:
numbers are NOT objects in the world — they are positions in a structure · 2 is just 'what comes after 1' · ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ — each layer was invented to solve the previous layer's limitations · ℝ is complete: every Cauchy sequence converges