# A Russell-like paradox **URL:** \ **Category:** Factual Questions\ **Created:** [August 15, 2002, 3:47pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032 "2002-08-15T15:47:57Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 3:47pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/1 "2002-08-15T15:47:57Z")
I’m told that if you start with x [symbol]Î[/symbol] y and y [symbol]Î[/symbol] x, you can derive a Russell-like paradox. I haven’t been able to figure out how, though. Does anyone know? ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 4:09pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/2 "2002-08-15T16:09:49Z")
John L. Kelley, in the appendix to his _General Topology_ derives the contradiction by appealing to the axiom of regularity ( if x is a non-empty set, then x has an element y disjoint from x), but this seems rather more technical than Russell’s paradox. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 4:16pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/3 "2002-08-15T16:16:58Z")
I’m not sure I see how that follows. Assume x = {y} and y = {x}. Then the conditions are met, and y is disjoint from x. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [August 15, 2002, 4:18pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/4 "2002-08-15T16:18:27Z")
For the sake of those of us with standards-compliant browsers, the symbol in question is the “is an element of” symbol, or ∈. I’m not as familiar with set theory as you might think; can you give an example of a Russel-type paradox? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 4:38pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/5 "2002-08-15T16:38:04Z")
Russell’s paradox is as follows: Let x be the set of all sets which do not contain themselves. If x contains itself, then x does not contain itself. Similarly, if x does not contain itself, then x contains itself. So x can’t exist. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 4:56pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/6 "2002-08-15T16:56:07Z")
Let z = {x,y}. By the axiom of regularity, z has an element disjoint from itself and plainly this element is either x or y. But x is not disjoint from z since y is an element of both and similarly y is not disjoint from z, as x is an element of both. Contradiction. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 5:11pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/7 "2002-08-15T17:11:24Z")
OK. That works. However, I’m pretty sure the book that I found this exercise in didn’t cover the axiom of regularity. I’ll check, though. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 5:20pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/8 "2002-08-15T17:20:31Z")
What axioms has the book covered? What is the subject-matter of the chapter in which the exercise appears? ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [August 15, 2002, 5:25pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/9 "2002-08-15T17:25:35Z")
Oooooooh… Neato. Sorry to hijack, but what’s the resolution to Russel’s Paradox? Or is it over my head? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 5:26pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/10 "2002-08-15T17:26:43Z")
It’s Ken Rosen’s _Discrete Mathematics and its Applications_. Very likely, he forgot that he hadn’t covered that particular axiom. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [August 15, 2002, 5:28pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/11 "2002-08-15T17:28:22Z")
Russell. I’ll get it right. Sorry. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 5:32pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/12 "2002-08-15T17:32:37Z")
> [@](#): > > \*Originally posted by Achernar \* > \*\*Oooooooh… Neato. Sorry to hijack, but what’s the resolution to Russel’s Paradox? Or is it over my head? \*\* The resolution is to disallow some collections of objects from being sets. There are several different ways to do that; the most popular is Zermelo-Frankel set theory. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 5:36pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/13 "2002-08-15T17:36:04Z")
Russell’s paradox implies that not all properties define sets ( roughly speaking, some things are too big to be sets), so you have to place some sort of restriction on what are allowed to be sets. A favourite approach of mine is Kelley-Morse Set Theory. Here we take “class” as a primitive notion and define a set to be a class which is a member of another class. We further stipulate u is an element of {x:…x…} if and only if u is a set and …u… ( Here the ellipsis indicates some condition on x). Now Russell’s paradox disappears. Define A = {x: x is not a member of x}. Then, in particular, A is an element of A if and only if A is a set and A is not an element of A. The upshot is that A is not a set ( it is a “proper class”). ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [August 15, 2002, 5:39pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/14 "2002-08-15T17:39:04Z")
Okay. So do Zermelo-Frankel and Kelley-Morse set theory resolve the paradox in the OP as well? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [August 15, 2002, 5:41pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/15 "2002-08-15T17:41:37Z")
If they both include the axiom of regularity. I hear it’s kinda controversial, though. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 15, 2002, 5:46pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/16 "2002-08-15T17:46:04Z")
But **Jabba** , can’t you then construct a similar paradox using “class” rather than “set”? I can re-define “set” to mean “banana” if I want, but that doesn’t really address the original paradox. ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [August 15, 2002, 5:49pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/17 "2002-08-15T17:49:40Z")
Aha, I see **ultrafilter**. From what I can tell of the [Axiom of Regularity](http://mathworld.wolfram.com/AxiomofFoundation.html), it basically says there exist no sets x, y, such that x contains y and y contains x. Am I right about this? I guess that’s a pretty good way to resolve a paradox; assume it’s irrelevant. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 5:52pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/18 "2002-08-15T17:52:28Z")
Michael D. Potter, in _Sets: An Introduction_ takes a different approach, based on the work of Dana Scott. Here, sets are built up stage by stage. Roughly, you start on day 0 with the individuals. The elements of each subsequent day are the individuals together with the elements and subcollections of the previous day. This avoids both paradoxes. For example, the situation in the OP is impossible because if x is a member of y then x must have formed on an earlier day than y and then y is not a member of x. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 6:02pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/19 "2002-08-15T18:02:55Z")
**Chronos** : I’m not sure I understand your point fully but it seems to rely on a Platonist view that we are trying to describe how sets “really” behave. However, this is not the case. The point of axiomatic set thory is that we are trying to retain as many of the results of naive set theory as possible while eliminating the obvious paradoxes. The three most common theories ( Zermelo-Fraenkel, von Neumann-Bernays-Goedel and Kelley-Morse) do this in different ways and it doesn’t matter ( from that point of view) which one you adopt. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [August 15, 2002, 7:22pm UTC](https://boards.straightdope.com/t/a-russell-like-paradox/124032/20 "2002-08-15T19:22:27Z")
> [@](#): > > \*Originally posted by Chronos \*]But **Jabba** , can’t you then construct a similar paradox using “class” rather than “set” Sorry, forgot this bit. No, you can’t. In KM, if x,y are classes with x a member of y and y a member of x then ( by definition of set) both x and y are sets and the proof I gave earlier applies. [Next page](https://boards.straightdope.com/t/a-russell-like-paradox/124032.md?page=2)