Here Professor Blass describes the following cumulative hierarchy of sets:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), then form all sets of these, then all sets whose elements are atoms or sets of atoms, etc. This "etc." means to build more and more levels of sets, where a set at any level has elements only from earlier levels (and the atoms constitute the lowest level). This iterative construction can be continued transfinitely, through arbitrarily long well-ordered sequences of levels. This so-called cumulative hierarchy is what I (and most set theorists) mean when we talk about sets.
We want to agree on the following principles:
- For every level there is a succeeding level.
- For every sequence of levels: $l_1,l_2,l_3,\dots$ there is a level succeeding all levels $l_1,l_2,l_3,\dots$. One might call this level "limit level".
Why is the axiom of replacement true under this interpretation of the term "set" (set = anything that is formed at some level of this hierarchy)?