We presented the *Zermelo-Fraenkel axioms* and derived some immediate consequences. In particular, we discussed Russell’s paradox and how it implies that the *universe * is not a set.

We briefly mentioned some alternatives such as Peter Aczel’s *antifoundation* axiom or Quine’s *New Foundations. *Aczel’s book can be found here. A nice short introduction to Quine’s NF can be found here.