From: gudeman@cs.arizona.edu (David Gudeman)
Newsgroups: sci.logic
Subject: Re: recursive definitions and paradoxes
Date: 23 Nov 92 20:22:51 GMT
Organization: U of Arizona CS Dept, Tucson

In article  <By6EKn.E5B@unx.sas.com> Gary Merrill writes:
]
]What I wonder mostly is whether what you are trying to do hasn't been
]done before, and with substantially more precision that your ideas
]currently have.

For a long time, I thought that my views on the paradoxes were the
common solution to the problem.  So I was somewhat surprised a while
ago to discover that I was considered a crank for refering to this
solution.  Since I don't really have a crank personality (you know,
the type that assumes that they are so brilliant that they can single
handedly solve problems that have eluded a large community for
centuries, and that everyone who doesn't see their solution is
stupid), I spent the next two years searching the literature for a
reference to this.

The solution was so obvious to me that I was certain it had been
proposed decades ago, but I never found a direct reference.  There are
some indirect references like "the lambda calculus avoids the
equivalent of Russel's paradox for functions", but no one seems to
have applied the work in this area back to set theory.  It still would
not surprise me at all to find a complete developement of these ideas
elsewhere, what surprises me is that there is so much resistence to
the idea.

I stumbled across the idea by imagining what would happen if you tried
to implement self-containing sets on a computer and then creating
Russell's paradoxical set.  The result would be a stack overflow due
to unbounded recursion.  I thought, "Oh, Russell's paradox is caused
by an ill-founded recursive definition."  End of query.  I didn't
bother to explore the situation any more until I first realized that
this is an unusual view.  Then I spent some time looking for where my
assumptions were going wrong so that I didn't have to have it pointed
out to me by someone else, but I never found any reason to change my
opinion.  Especially given that most of the other solutions are
demonstrably wrong.

That is why I brought the idea up in this newsgroup.  I was hoping
that someone would either provide the references I could not find, or
would be able to provide some sort of argument against my solution.
So far, I've seen neither one.  This discussion of terminology, while
it may be necessary for communication, is not enlightening.  No one
has responeded in a substantive way to my contention that I can
express Russell's paradox _without_ resorting to a universal set.  Or
to my contention that I can express the liar's paradox _without_
refering to sentences in a language.  Or to my contention that all of
the paradoxes can be unified with one explanation.  These are the
interesting points.  The definitions are just a vehicle.

]This appears to be the introduction of yet more unexplained notation
]in order to explain the previous set of unexplained notation.

No, I explained the previous notation before I introduced this
extension.  I gave explicit rules for introducing definitions and for
using them.  What more do you want?

]I take it that you don't want a definition to, by itself, imply
]the existence of the thing defined.

Actually I would, if there were some way to make sure this does not
lead to an inconsistency.  I tried to accomplish this with the axiom
of definition, but as someone else has pointed out, this makes the
definition eliminable.
-- 
					David Gudeman
gudeman@cs.arizona.edu
Newsgroups: sci.logic