Re: Cantor Confusion

In article <1160302222.613036.300930@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> Han.deBruijn@xxxxxxxxxxxxxx writes:
*** T. Winter wrote:

I would say that all forms of mathematics are grounded on axioms (or dogmas
as you prefer to say). But contrary to dogmas, axioms can be negated to
get another form of mathematics. Dogmas are absolute truths, axioms are
only absolute truths within some realm of discourse.

If it is so simple, where then come these heated debates (about the
Balls in a Vase at noon) come from?

Because some of the people in the discussion only use intuition, and not
provable concept, but they keep stating that there intuition tells them
that there are contradictions, without giving proof of any of this.

If you are, say, discussing in a context where the axiom of infinity is
assumed, you can not get a contradiction when during your proof you, at
one stage or another, use the contradiction of that axiom.

The balls in vase problem suffers because the problem is not well-defined.
Most people in the discussion assume some implicit definitions, well that
does not work as other people assume other definitions. How do you
*define* the number of balls at noon? You can not use limits, because the
limit does not exist when you use standard mathematics. So using standard
definitions there is no answer. More precise, given the sequence of sets:
{1, ..., 10)
{2, ..., 20}
{3, ..., 30}
etc., is there a limit? Well, no, there is no defined limit unless you
define what a limit of sets looks like. I have never seen a definition
that tells me how the limit of a sequence is defined. The limit of the
size of the sets also gives no answer, because that limit does not exist.
Strange enough, when somebody goes on to define things, *you* question his
definitions, rather than the result.

And why then are some axiom systems
so much more dominant than others?

That is easy. Some axiom systems give results easier than others. For
instance, the axiom of infinity asserts that the set of natural numbers
does exist. This means a simple definition of limits, also this means
that the definitions of the reals in their various ways work,
differentiation and integration get properly defined, and in the end, even
things like eigenvalues of matrices are properly defined.

To build the same without the axiom of infinity may be possible, but to me
it does not look as being exactly easy.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.