First is the axiom of extensionality, if two sets contain exactly the same elements, then they are the same set. The second one is axiom of empty set. Third, axiom of pairing, if you have two sets, you can put them together into a new set.

axiom of unition meaning to merge sets together, axiom of power set, which is the set of all sets, axiom of infinity ,like when we count number it can go forever, the seventh is the axiom schema of replacement, making a new set by changing elements, it’s like starting with {1, 2, 3} then apply a rule of doubling you get {2, 4, 6}, number eight is the axiom of regularity, meaning a set can not contain itself, and there are no infinite loops of sets inside sets. Lastly axiom schema of specification, given a set you can create a new set by picking only the elements that satisfy some condition.

These axioms are the basic rules that define what a set can do and how sets interact. Everything in math, from numbers to geometry to functions, is built on top of these ideas!