First-Order Language of Arithmetic Explained ✏️

The first-order language of arithmetic is austere: it contains only the constants 0, 1, the relation symbols =, ⟨, and the function symbols + and ×. Why so austere? Because we can construct any arithmetic expression out of just these. So we can construct relatively simple proofs, using basic constants, functions, etc., which will hold for numbers we will never actually see. And for classical arithmetic, that's good news.