Re: Axiomatic set theory is still contradictory

On Feb 15, 4:09 am, David C. Ullrich <ullr...@xxxxxxxxxxxxxxxx> wrote:
On 14 Feb 2007 22:13:17 -0800, victo...@xxxxxxx wrote:





In his initial formulation of the axioms of set theory Ernst Zermelo
used the vague notion of a definite property to define his new axiom
of substitution. Fraenkel and others improved this by introducing the
formal concept of well formed formulae to the basic terminology of
axiomatic set theory.

Well formed formulae are variables concatenated by very specific rules
of formation in order to represent valid predicates in set theory. But
is the idea of a variable itself well formed? It appears not. The
very same paradox that confounded naïve set theory can be shown to
dissemble the concept of a variable.

Let us understand by a variable a symbolic representation of an
object. Thus x could represent a set, a natural number, a proposition
or any other entity associated with an assigned concept.

Sadly, variables are themselves objects. The collapse begins. Some
variables - metavariables - will be defined that represent other
variables. Thus N can be defined to represent all those variables
that represent natural numbers.

In which case some metavariables will include themselves in the
variables they represent while some will not. Thus, the metavariable
R that represents all Roman alphabet variables includes itself, while
the metavariable G that represents all Greek alphabet variables does
not.

Um. It's simply not true that a variable represents exactly
one thing, in the way you suggest. For example, saying that
R represents the Roman alphabet is false. We _could_ _say_
that R represents the Roman alphabet, but unless we
_assume_ that it does so it doesn't. It could just as
well represent the set of real numbers, for example.

That's all speaking about the informal uses of these notions.
Luckily in a _formal language_ variables do not in fact
"represent" anything at all, so all of this has no possible
bearing on the consistency of any given formal system.

Consider now the variable y that represents all the variables that do
not represent themselves.

If there _were_ such a variable you'd have a point.

Perhaps you meant to say "Let y represent the set of all
variables that do not represent themselves". Fine, except
that there's simply no such thing as a variable that
represents itself (or not) - see above.

In order to make sense of the "problem" we need to
_first_ assume that we've _fixed_ an interpretation
for every variable in the universe. But once we've
done that we can no longer say "let y be a variable
that represents such and so", because y _is_ a
variable, and it may well _not_ represent what we say.

Does y represent itself? If it does it
doesn't and if it doesn't it does. We've seen it all before only this
time it applies not to propositions or to sets but to a concept even
more basic.

We cannot escape the conclusion that the formula 'x e y iff x e z',
for example, which although being well-formed, meaningful and possibly
true in axiomatic set theoy, makes use of the ILL-formed variable y.
Axiomatic Set Theory has no way of blocking this miscreant variable
and must live with the discomfort that even if internally consistent
it cannot escape a link with external contradiction.

************************

David C. Ullrich- Hide quoted text -

- Show quoted text -

1. Within the context of discourse a variable can only represent one
thing.
y represents A
y represents B
A is not B
is just not possible within a specific context of discourse. Within
any given context of discourse we can define any (but only one)
variable to represent all variables that do not represent themselves.

2. The set of all variables would be attacked by some as not being
part of (some) axiomatic set theory

3. If variables do not represent anything how do we know that they are
set theoretical and not abstract algebra? How do we know that the
formulae are true of objects of one domain and not another? How do we
know that the axioms of set theory apply to sets and not natural
numbers or pink elephants?


.

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