# Limiting set theory using symmetry

[Cross-posted from here]

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?

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I don't know the paper you're interested in, and so I can't assess it. But in general you have to be a little cautious about papers on the arxiv, just like you have to be cautious at a flea market. I would view a paper introducing a new type of set theory with a little skepticism at first. That being said, the main alternative system for set theory is "New foundations" (NF); you can find a list of references on its Wikipedia article. –  Carl Mummert Nov 13 '10 at 14:10
@Carl: Thanks for the pointers. The reason I am interested in that paper is as corroboration for a line of thought I am pursuing at the moment. So while I will approach the paper with due skepticism, I am looking for a certain idea which is likely to be pursued in that line/paper. –  Muhammad Alkarouri Nov 13 '10 at 14:31
Perhaps the OP is asking about symmetry models? That is, models constructed either from urelements by respecting a certain group of symmetries of the urelements? Or the forcing models for $\neg AC$, which are build by using symmetric names? These have little to do with resolving Russell's paradox, though. –  Joel David Hamkins Nov 13 '10 at 22:27
@malkarouri: Any chance this is what you were thinking of? mathoverflow.net/questions/49721/… –  Andres Caicedo Dec 17 '10 at 20:16
Does the poster mean something like Holmes's "Symmetry as a Criterion for Comprehension Motivating Quine's 'New Foundations'"? –  Malice Vidrine Aug 13 '13 at 7:04

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.)

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.
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 $y$.
Is this what you mean by symmetry condition?

Unfortunately, it is not known whether the consistency of NF follows from any theory that is believed to be consistent.

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If you're sure the paper in mind was on the arxiv, then this paper of Harvey Friedman's 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 principle of symmetric arguments could be of use to you nonetheless.

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