Re: virus: Quotes
Sat, 26 Oct 1996 18:50:51 -0500 (CDT)

On Sat, 26 Oct 1996, Peter =?iso-8859-1?Q?=D6kner?= wrote:

> KMO wrote:
> >Don't sweat it. I'm about to do likewise, and I'm not even going to
> >provide any analysis. (...)
> Not even when you are being contradictory to yourself? Or am I missing
> something?
> Kuhn's cool and I liked the modell of usefull maps, but than this;
> >(...) if the statement "All truth is relative" is
> >objectively true, it's objectively false.(...)
> Are you starting up the merry dance of Russels Paradox again - that of a
> statement about sets of sub-sets that does not include itself - or are you
> testing us, or what? Curioser and curioser!

No. This is basic contradiction reasoning. In classical logic, this

If A, then NOT-A. [technical! This isn't "if and only if"]

Presumed data: NOT-A
What do I get [contrapositive]: NOT-A. [I'm OK.]

Presumed data: A
What do I get [straight]: A, NOT-A. [This blows up in classical logic.
It does NOT blow up in intuitionistic logic.]

THUS, the only presumed data that does not self-destruct is NOT-A.

The quantifers do not save us here; the predicate "is relative" has only
one argument, which makes the thing effectively decidable [if not
practically; technical usage]. This system is exempt from Godel's
Theorem; it's too simple.


Ok...Time to sketch the setup Russell's Paradox, and the nominal escape.

Take the following as undefined:
True, and False
Enough connectives to get Propositional Calculus up.
ONE binary relation, loosely called "in". [There's a formal notation,
but it just doesn't fit into ASCII.]

The following is somewhat recursive. I have not sequenced it correctly;
think of it as a set of parallel definitions. I'm summarizing the
important points, not actually how one constructs them.

A *class* is what belongs on the RHS of "in".
A *set* is what belongs on the LHS if "in".
A set is actually defined to be a class such that one can decide which sets
are members of it.

The first thing one does, when going into gory technical things here, is
demonstrate that the null/empty set exists. It is unique, and has
exactly 0 members. It is easy to decide which sets are members of the
empty set. None of them.

There are ten or so rules about how to construct sets. These boil down
to writing computable rules for determining which sets are elements of
your would-be set. A class which does not satisfy these rules is called
a proper class. A class which does satisfy these rules is called a set.

Russell's paradox results from trying to computably decide membership in
a proper class. This is defined out of existece in the above schema.

/ Towards the conversion of data into information....
/ Kenneth Boyd