Re: Cantor Confusion

mueckenh@xxxxxxxxxxxxxxxxx wrote:
*** T. Winter schrieb:

In article <1164816790.379338.139370@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> MoeBlee schrieb:
> > > I think if one swells to explosion about his knowledge of set theory,
> > > he should at least know the very foundation. But I know, that you do
> > > not even understand the simple texts of Fraenkel et al.
> >
> > No, YOU radically MISunderstand what Fraenkel, Bar-Hillel, and Levy
> > wrote.
>
> How can you judge about that without the slightest idea of what they
> wrote? Only for the lurkers: Fraenkel et al. write: "Platonistic point
> of view is to look at the universe of all sets not as a fixed entity
> but as an entity capable of "growing", i.e., we are able to "produce"
> bigger and bigger sets." So a set (like the set of all sets) is not a
> fixed entity.

There is nothing in that that shows that a set is not a fixed entity.

The universe of all sets can grow. Define: "The universe of all sets is
called the set of all sets", and you see it.

As to what Fraenkel, Bar-Hillel, and Levy wrote, they are underscoring
the fact that different axioms yield different universes of sets. That
is what they mean by the universe of sets "growing" (scare quotes in
original text).

You are able to produce bigger and bigger sets, but they are all
different.

That is a matter of definition. If you consider a fixed set then it is
fixed. Small wonder. If you consider a variable set then it is
variable and perhaps changes its cardinal number.

You can go ahead and provide some theory in which there are such
variable sets. Meanwhile, there is nothing like it in Z set theories.
And Fraenkel, Bar-Hillel, and Levy's comments do not contradict this.

An easy example which
should not escape you: The set of states of the EC has been growing and
probably will continue to grow.

Which you'll have a hard time proving to be a set in Z set theory.

Of course no one denies that the everyday, non-mathematical, sense of
'set' includes the idea that many everyday conceived sets have changing
membership. But that is distinct from the mathematics of Z set theory,
which has an axiom that determines that the sense as applied to such
theories is distinct from the everyday sense. No one is stopping anyone
from formulating a mathematical theory in which sets have changing
membership; but the option of doing this does not change that in Z set
theories, sets do not have changing membership.

When the universe has grown it allows bigger sets than
where originally allowed, but the sets originally allowed are still
sets and still the same and did not grow. How you conclude from the
above statement that sets themselves are growing escapes me.

It is simply a matter of definition.

No, it's a matter of an axiom.

MoeBlee

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