Some time ago I enrolled in a class called "Introduction to Real Analysis" in which we learned a lot about Set theory and it's axioms, I learned about the specification axiom, about Russell's paradox and (my favorite part) the nonexistence of the set of all sets. Later on discussing with friends they told me that this was nonsensical since the axiom of regularity (which I didn't knew) solves the Russell's paradox.
Reading in Wikipedia and in other materials about the axiom of regularity I couldn't understand why it is needed. It isn't clear to me how it is true and why we should 'believe' in it, Isn't the axiom of specification more than enough to define a set? Aren't the two getting in the way of each other?
And is my favorite proposition false? The universe exists?