Universal set--is its definition circular? Consider NFU set theory as presented in this:
http://math.boisestate.edu/~holmes/holmes/head.pdf
On page 15 of that pdf it states that the following is an axiom:
The set $\{X\colon X=X\}$ exists.
Let $V = \{X\colon X=X\}$.
$V$ is an element of $V$ as $V=V$.
How is this not circular?
 A: While we are very much used to this, and it seems very natural to us, the axiom of foundation which asserts that $\in$ is well founded (every set has a $\in$-minimal element) is independent of the rest of the axioms.
It is a very interesting construction to see how one can have a model of set theory in which there is some $x=\{x\}$. It can get stretched even further, $P(x)\in x$ for some $x$ (where $P(x)$ denotes the power set).
Going through the .pdf that you linked I can observe several things:


*

*There is no axiom of foundations in NFU (somewhat ironic, isn't it?). Therefore there is no real contradiction in $V\in V$.

*In chapter 7 (page 39) it is proved that $\in$ is not a set-relation of the universe, however $\subseteq$ is. Meaning that we care more about subsets than we do about elements.

*There is no "axiom of power set" either, so $P(V)$ need not exist as a set, which is just fine with the above, although as I said earlier, it is quite possible to have $P(x)\in x$.

*Comprehension itself is bounded. Unlike ZFC, however, the bound is not by taking a subset of some pre-existing set, but rather bounding the formula we use to a stratified formula - the definition of which appearing on page 40 of the .pdf file.
I'd guess that the further you get in this book, the more observations you can collect. However skimming through the first 50 would seem enough to gather the above and see that there is no real apparent paradox in NFU. It is only your intuition which was heavily laced with ZFC which stands in your way.
