Re: Cantor Confusion

In article <1164878462.838895.276420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
*** T. Winter schrieb:
....
> 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.

And that is the confusion. If it is a set it cannot grow, but as Fraenkel
et al. do not define it as a set it is allowed to grow. They do not state
that a set can grow because they do not state that the universe is a set.

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

That is not "the set of states". You can talk about "the current set of
states" or about "the set of states in 1957" or whatever. At least
mathematically. In mathematics, by definition, a set can not grow.
You are, of course, entitled to use another definition, but that will
not clarify the discussion at all (and you are not using standard set
theory).

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.

Yes, with your definition a set can grow, but you put yourself outside
set theory, and you must at first consider all results from set theory
unproven theorems in your theorem, and you need to prove them (if
possible).
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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