Limiting set theory using symmetry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:01:51Z http://mathoverflow.net/feeds/question/45918 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45918/limiting-set-theory-using-symmetry Limiting set theory using symmetry malkarouri 2010-11-13T13:49:12Z 2010-11-28T04:22:13Z <p>[Cross-posted from <a href="http://math.stackexchange.com/questions/10086/limiting-set-theory-using-symmetry" rel="nofollow">here</a>]</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/45918/limiting-set-theory-using-symmetry/45970#45970 Answer by Stefan Geschke for Limiting set theory using symmetry Stefan Geschke 2010-11-13T20:57:19Z 2010-11-13T20:57:19Z <p>Both type theory and New Foundations (which is inspired by type theory) use syntactic rather than size restrictions in the formation of sets. (New Foundations is mentioned in Carl Mummert's comment.) </p> <p>Russell's paradox practically rules out unrestricted comprehension, i.e., the collection of all sets with a certain property cannot be a set for all properties.<br> In New Foundations you can form the set of all sets with a property that can be described in a specific syntactical form. The point here is that constructs like $x\in x$ are forbidden. You have to be able to assign a natural number to every variable in the formula describing the property in question such that whenever "$x\in y$" appears in the formula, then the number assigned to $x$ has to be strictly less than the number assigned to $y$.<br> Is this what you mean by symmetry condition?</p> <p>Unfortunately, it is not known whether the consistency of NF follows from any theory that is believed to be consistent. </p> http://mathoverflow.net/questions/45918/limiting-set-theory-using-symmetry/46013#46013 Answer by Ed Dean for Limiting set theory using symmetry Ed Dean 2010-11-14T04:00:54Z 2010-11-14T04:00:54Z <p>If you're sure the paper in mind was on the arxiv, then <a href="http://www.math.ohio-state.edu/~friedman/doc/8-acompthyevery5_16_99.doc" rel="nofollow">this paper of Harvey Friedman's</a> isn't it. But since you're after "corroboration for a line of thought [you are] pursuing at the moment," maybe it and its treatment of a <em>principle of symmetric arguments</em> could be of use to you nonetheless.</p>