We like having small sets of primitive notions ("exists", "for all", "set"). The reason we have words like "surjection" is to abbreviate things which are otherw

We like having small sets of primitive notions ("exists", "for all", "set"). The reason we have words like "surjection" is to abbreviate things which are otherwise unwieldy to state precisely using our small set of primitives. It's analogous to the reason we might program in high-level languages and then compile down to machine code: because the low-level code is much easier to execute directly (analogously, the primitives "forall", "exists", "set" are extremely intuitive and we're nearly all sure that they are non-contradictory).