# In search of a set theory with specific properties

I'm in search of a set theory that satisfies the following requirements.

1. There is a universal set $V$ such that $\forall x(x \in V)$. So for example, $V \in V$.
2. Sets whose elements are 'large' exist. e.g. I want $\lbrace V,\emptyset\rbrace$ to be a well-defined set with cardinality $2$.
3.  Sets form a boolean algebra; in particular, the complement of a set always exists.
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Positive set theory (en.wikipedia.org/wiki/Positive_set_theory) seems to have these properties –  Trevor Wilson Sep 1 '12 at 13:58
He Trevor, I added a third requirement on the basis of your comment. Do you know if there is a positive set theory satisfying this property? I can't seem to work it out from the wikipedia page. –  user18921 Sep 1 '12 at 14:23
New Foundations en.wikipedia.org/wiki/New_foundations ? –  Ben Crowell Sep 1 '12 at 14:36
Thanks, NF does indeed satisfy all of the above properties. So too does NFU, which according to Randall Holmes does not suffer the major objections to NF. I have just downloaded, and are currently reading, Randall's free ebook "Elementary Set Theory with a Universal Set". –  user18921 Sep 1 '12 at 15:12
Perhaps these comments are better posted as answers? The site seems to work better that way. –  Joel David Hamkins Sep 2 '12 at 0:11
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