🚀 go-pugleaf

In article <Oct05.201328.96457@yuma.ACNS.ColoState.EDU> sa114984@longs.lance.colostate.edu
(Steven Arnold) writes:

>	I am not an expert in set theory or the predicate
>calculus, but I was leafing through a book on the subject and the
>following informal "proof" occured to me.  Could any of you experts out
>there tell me if this contains a major flaw -- and if so, what it
>is?  Thanks.
>
>	The proposition to be proven is: A set can never contain
>itself as a member.
>	Suppose X is an arbitrary set which contains itself as a
>member.  All other members within the set can be symbolized as
>'g'; therefore,
>
>		X = {g,X}		(1)
>
>(Of course, g may be the empty set.)
>	Eq. 1 can be expanded as follows:
>
>		X = {g, X'}
>		        /\
>		       {g,X''}
>		          /\
>			 {g,X'''}
>
>...and so on forever.  (The ' symbol is merely meant to
>distinguish an X of one iteration from that of another.)  X'',
>for example, is equivalent to {g,X'''}; X''' is equivalent to
>{g,X''''}, etc etc.
>	The number of X's below X'' is infinite; nonetheless, it
>is certainly true that X' contains X'' as well as all the X's
>contained by X''; therefore X' contains one more X than X''!  It
>follows that X' != X'', for X' contains an X not contained in X''. 
>By the same reasoning, X != X', because X contains one more X
>than does X'.  Therefore, for any arbitrary set X, X cannot
>contain itself as a member.
>
>Steve

What you have just described is, I believe, what is called the
`unfolding' of the set x={x}.  It is a typical non-well-founded set.
A _non-well-founded set_ is a set that need not have an epsilon-minimal
element.  There will exist an infinite "descending" chain of the
epsilon-relation.  This is why it is called a non-well-founded set.  It
is not built out of pre-existing elements. The set x={x} can also be
represented by an infinite tree with a set assigned to each node.  
It would look like this, with an arrow indicating membership.

x-->x-->x-->x-->...

The existence of non-well-founded sets can be made into an axiom, the
Anti Foundation Axiom (AFA).  The resulting theory is very interesting
mathematically and has practical and theoretical applications.  See

Peter Aczel, _Non-Well-Founded Sets_.  CSLI Lecture Notes 14, (1988).

-- 
Marko Amnell
amnell@klaava.helsinki.fi
Graduate Student in Philosophy