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From: robertj@garfield.UUCP
Newsgroups: sci.math,sci.math.symbolic,sci.philosophy.tech
Subject: Re: Russell's set of sets which... paradox
Message-ID: <3830@garfield.UUCP>
Date: Mon, 27-Jul-87 09:18:41 EDT
Article-I.D.: garfield.3830
Posted: Mon Jul 27 09:18:41 1987
Date-Received: Wed, 29-Jul-87 02:30:29 EDT
References: <1214@utx1.UUCP> <6678@reed.UUCP> <744@cadnetix.UUCP>
Reply-To: robertj@garfield.UUCP (Robert Janes)
Organization: Memorial University of Nfld, St. John's
Lines: 56
Keywords: types,classes,Principia
Xref: utgpu sci.math:1566 sci.math.symbolic:98 sci.philosophy.tech:294
Summary: There are other ways than Frankel-Zermelo

In article <744@cadnetix.UUCP> pem@cadnetix.UUCP (Paul Meyer) writes:
>[]
>	Actually, the paradox as stated leads to a very significant con-
>   clusion:  not everything is a set.  Modern mathematics (at least as I
>   was taught it) draws a distinction between "classes" (things which
>   follow the intuitive idea that a set can be anything) and "sets" (which
>   obey certain rules).


Several people have pointed out the simplest (I think) solution found to the 
Russell Paradox, that is that the object that Russell is talking about is not
a set. The version of an axiomatization which arrives at this conclusion
with which I am aquainted is the Frankel-Zermelo Axiomatization in which
class and membership are the primitive (undefined) concepts (not sets!) and
a set is defined as a class which is a *member* of another class. Classes
which are not members of other classes are not sets and we cannot talk
about their "power class" and so forth, they are proper classes. The object
Russell defined, that is the class of all sets not members of themselves,
can easily then be shown to be one of these proper classes as assuming it
to be a set leads to a contradiction, that is the Russel Paradox.

	In all fairness however it should be pointed out that Russell in
conjunction with Whitehead came up with another solution to the problem
which does not invlove just saying that the Russell Class (R) is not a set.
Russel observed that the Russell Paradox in some way arose because of
mixing levels. That is we have here a sentence that talks about sets
of seemingly simple things (eg the set of orange cats) and sets that
talk only about other sets (eg R itself). What Russell and Whitehead
did (more or less) was say that there is a hierarchy of sets, the simplest
being type zero sets which in some way correspond to things like the
set of all grey dogs and then moving on to type one sets, which deal
with type zero sets and so forth. This was called the Theory of Types.
The key point in this theory is that one could not talk about type zero
sets interchangeably with type one sets. Thus the equation $X in R$ is
ok if $X$ runs over the type zero sets but is meaningless if we put
a type one object in the place of $X$. Thus Russell resolves the paradox
by banishing the sentence $R in R$ rather than by banishing $R$.

	This description may be wrong in details but the essential idea
is there. The resolution of set theory into an infinitely ascending
hierarchy of types which cannot be arbitrarily mixed in logical statements.
As can easily be imagined this becomes quite messy eventually particularly
when compared with F-Z which has only two levels, class and set. The system
outlined by Russell and Whitehead's Principia Mathematica (which every
wants to have read but no-one has, a classic) does capture everything
that F-Z does however and I believe Godel's work was actually written
with the Principia in mind as his model of a formal system. I think it
can be fairly safely said that very few and probably no working 
Mathematician or Logician uses the system set out in Principia. 

	Lord Russell said that after he noticed his paradox he spent
ten years looking at a blank sheet of paper, then wrote Principia
and then never did another good piece of Mathematics (or logic) again.
At least he did propose a workable if cumbersome solution.

					Robert Janes