From: owner-standard-upper-ontology@LISTSERV.IEEE.ORG 
[mailto:owner-standard-upper-ontology@LISTSERV.IEEE.ORG] On 
Behalf Of John F. Sowa
Sent: Monday, April 04, 2005 8:18 AM
To: 'SUO WG'
Subject: Re: How about Self-Inconsistency

There was an offline discussion about how Tarski's
hierarchy of metalanguages would avoid the paradox
of the lying Cretans and/or set membership.

Following is my response.

John Sowa
____________________________________________________

Tarski method of avoiding the liar paradox was to
introduce a hierarchy of languages:

    L0, L1, L2, ...

The language L0 could only refer to entities in the
base universe U.  L1 could refer to individuals in U
or to statements in L0.  In general, any language Ln
could refer to U and to statements in any Li where i<n.

By insisting on this hierarchy, Tarski would make
any sentence of the following form illegal:

    This sentence is false.

That sentence would have to occur at some level Ln,
and it refers to a statement (itself) at level Ln.
The paradox of the lying Cretans involves several
statements that would have to be at the same level.

Tarski's approach does not solve the problem about
set membership because there is no need for metalanguage.
For example, you can talk about the set of all sets that
are members of themselves with an expression such as

    {x | memberOf(x,x)}

There is no contradiction here.  If you want to express
the set of all sets that are not members of themselves,
you get

    {x | ~memberOf(x,x)}

This expression is not contradictory by itself.  The
contradiction arises with another axiom that various
people, starting with Cantor, wanted to assume:

    For every monadic predicate P, there exists the
    set {x | P(x)} of all x's for which P(x) is true.

If P is defined as (lambda x)(~memberOf(x,x)), this
axiom creates a contradiction.

There are several ways of getting around this, either
by putting restrictions on what it means to be a set
or by restricting what kinds of predicates are allowed
in expressions of the form {x | P(x)}.

Frege got into trouble because he didn't assume any
restrictions on his axioms, but Zermelo saw the problem
and found a way to get around it.  Since Zermelo didn't
advertise the solution as loudly as Russell did, the
paradox is called Russell's paradox, not Zermelo's.




 


-----Original Message-----
From: owner-standard-upper-ontology@LISTSERV.IEEE.ORG 
[mailto:owner-standard-upper-ontology@LISTSERV.IEEE.ORG] On 
Behalf Of John F. Sowa
Sent: Monday, April 04, 2005 1:33 PM
To: Rich Cooper
Cc: 'SUO WG'
Subject: Re: How about Self-Inconsistency

Rich,

I'm glad that you like Tarski's hierarchy because
it happens to be included in my paper on Laws, Facts,
and Contexts (as well as Ch. 5 of my KR book).

For more about his hierarchy (with references), see
Section 4 of the following paper:

    http://www.jfsowa.com/pubs/laws.htm
    Laws, Facts, and Contexts

I also had another offline question about mereology
and whether it can avoid the paradoxes about sets.
My response is below.

John
_____________________________________________________

The partOf relation of mereology corresponds to the
subset relation rather than the membership relation.
Every set is a subset of itself, and in mereology,
every collection, fusion, sum, or whatever you call
it is a part of itself.

There is no paradox with the set of all sets that
are not subsets of themselves because that set is
empty -- every set (including the empty set) is
a subset of itself.

In set theory, the assumption of two notions of
containment -- subset and membership -- is very
confusing to beginning students.  That is one
reason why mereology is a better model of plurals
in natural language than set theory, and that is
also why I think that the attempt to reduce all
of mathematics to set theory is misguided.

I would prefer to use mereology as the theory of
collections.  However, mereology is too weak to
create new things out of the empty set -- you need
to assume something else in addition.  My preferred
foundation for mathematics would be the integers
plus mereology.

Many mathematicians share the opinion that the
foundations of arithmetic are far more solid than
the foundations of any version of set theory.

The fact that there happens to be many different
axiomatizations of set theory does not give any
confidence that any of them are suitably secure
to serve as the foundation for all of mathematics.
In fact, most working mathematicians seriously
doubt that *any* foundations are necessary -- they
generally ignore all of the stuff the logicians do.