Russell's Paradox

Russell's Paradox is a headache of logic and naive set theory.

Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. Discovered by Bertrand Russell in 1901, the paradox has prompted much work in logic, set theory and the philosophy and foundations of mathematics.

To understand that example (which made my head spin for a few seconds), you have to take everything completely literally. "The set of all teacups" is literally referring to all of the teacups in the universe. A set can contain anything, however you define it, including other sets. That set is not a member of itself because a set of teacups is not a teacup, so doesn't fit the set's definition.

The "set of all non-teacups" is just as literal. This is referring to everything, everywhere, that isn't a teacup. That set includes all sets, also, since a set is not a teacup.

Since it includes all sets, "the set of all non-teacups" is a member of itself.

That's all to explain the types of sets we're talking about, and how sets can contain other sets. Now imagine a set that contains only sets that are not members of themselves. (Like the first set of teacups.) This new set is the paradox. If that new set is a member of itself, then it is excluded from being a member of itself by the very rule that makes the set. Yet, if it's not a member of itself, then it meets the rule that defines the set and becomes a member of itself...

Ow. Now you see where the headache comes from. I love this stuff!

Incidentally, there is no official solution to this paradox. What is commonly done now is to limit the creation of sets to those which don't send logicians into infinite loops. (They now use "ZF", the axiomization of set theory.)

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RE: Russell's Paradox

At 02:13 PM 11/4/2002 -0500, Brian A. wrote:

>All it indicates is that our language is able to express things that don't
>make sense -- I doubt that this is news to anyone. Some disciplines have
>formal names for this idea... in mathematics there's the concept of a
>well-formulated function or well-formulated question.

This is definitely *not* an issue of whether or not a given idea can be
expressed in our language. The idea of a set of all sets that are not
members of themselves is a useful conceptualization of the paradox, but the
paradox goes much deeper. As Seth noted, the problem was that naive set
theory did not set adequate conditions on defining sets -- under the ZF
axiomatic system, the set of sets that do not contain themselves as an
element is not a valid set.

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Re: Russel's Paradox

At 01:07 PM 11/4/2002 -0500, Seth wrote:

>Incidentally, there is no official solution to this paradox. What is
>commonly done now is to limit the creation of sets to those which don't
>send logicians into infinite loops. (They now use "ZF", the axiomization
>of set theory.)

Right because Russell proved that naive set theory was inconsistent and
that it was therefore possible to derive contradictions from it (such as
the set of sets that do not contain themselves as a member), and, of
course, once you've got a contradiction you can prove any arbitrary
proposition. It's kinda weird to think that this simple paradox essentially
rendered naive set theory useless.

It's interesting that Russell's solution to the paradox (to create a
hiearchy of types of propositions) was quickly rejected.

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