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Re: An element that contains itself

• From: Shaun McCance <shaunm@gnome.org>
• To: xml-dev@lists.xml.org
• Date: Wed, 06 Aug 2014 13:31:25 -0400

On Sun, 2014-08-03 at 19:09 +0000, Costello, Roger L. wrote:
> Long ago (in the last half of the nineteenth century) some very smart
> people sought to build a solid foundation for mathematics. They
> decided to use sets as the foundation. They defined a set as:
> 
> 	A set is a collection of distinct objects.
> 
> However, with that definition a set can contain itself, just like this
> entity contains itself: 
> 
> <!ENTITY Set "&A; &B; &Set;">
> 
> A fellow by the name of Bertrand Russell realized the problem with
> allowing a set to contain itself (the problem he identified became
> known as Russell's Paradox), so the mathematicians changed the
> definition of set to this:
> 
> 	A set of a collection of distinct objects, 
> 	none of which is the set itself.

Sorry, this just isn't true. Set are allowed to contain themselves in
every formulation of set theory I've ever seen. Mathematicians have no
problem whatsoever with the idea that a set contains itself.

Russell's paradox concerns defining a set to contain all sets that do
not contain themselves. Does that set contain itself? The paradox is
solved in ZFC, the most common axiomatization of set theory, basically
by not assuming that a set must exist just because your language allows
you to describe it.

--
Shaun

• Follow-Ups:
  • Re: An element that contains itself
    • From: David Rudel <fwqhgads@gmail.com>

• References:
  • An element that contains itself
    • From: "Costello, Roger L." <costello@mitre.org>