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
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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.