  In this paper, a generalized version of the von Neumann universe known as the
total universe is proposed to formally introduce non-well-founded sets that
include infinitons, semi-infinitons and quasi-infinitons in Russell's paradox.
All three infinitons are part of infinitely generated sets that are generators
of non-well-founded sets. The total universe combines the well-founded sets
with the non-well-founded ones and turns out to be a model of ZF minus the
axiom of regularity, which can be shown invalid in defining the well-founded
sets. Also, the total universe is shown to be free of Russell's paradox.
