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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. [Edit] 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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up vote 5 down vote accepted

Here is a CW answer incorporating the answers in the comments. Both NF (New Foundations) and NFU (New Foundations with urelements, obtained from NF by weakening the axiom of extensionality) satisfy conditions (1) through (3).

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