Asaf Karagila
I don't have much choice...

## In praise of Replacement

I have often seen people complain about Replacement axioms. For example, this MathOverflow question, or this one, or that one, and also this one. This technical-looking schema of axioms state that if $\varphi$ defines a function on a set $x$, then the image of $x$ under that function is a set. And this axiom schema is a powerhouse! It is one of the three component that give $\ZF$ its power (the others being power set and infinity, of course).

You'd think that people in category theory would like it, from a foundational point of view, it literally tells you that functions exists if you can define them! And category theory is all about the functions (yes, I know it's not, but I'm trying to make a point). Continue reading...