If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in terms of size. For example, the set of all sets must be excluded.
I recall that I came across a paper in Arxiv some time ago which explained that a useful restriction may be expressed in terms of symmetry conditions rather than size. Can anyone explain to me the concept and/or provide a link to the paper in question?
I don’t know of any set theories using symmetry to avoid the paradoxes, but there are a number of counterexamples to your claim that “the set of all sets must be excluded.” The most famous is Quine’s New Foundations; my preferred theory is Church’s (first) Set Theory with a Universal Set (presented 1971, published 1974). For more details, see the Wikipedia article on Universal Set (Disclaimer: I started the article. I don’t know of any published accounts of Church’s later theory(s), though I discuss them in my forthcoming rewritten doctoral thesis.)
