Re: Cantor Confusion

In article <1164974976.838760.48250@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
*** T. Winter schrieb:
In article <1164878462.838895.276420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
....
> 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 talk about the set of all sets, but cannot use it for the well
known reasons.

No, they do not talk about the set of all sets. That *you* define it as
such does not mean that they are talking about that.

They do not state
that a set can grow because they do not state that the universe is a set.

Here is a book on ZF set theory: Karel Hrbacek and Thomas Jech:
"Introduction to Set Theory"

What is the relation with what Faenkel et al. did write?

Marcel Dekker Inc., New York, 1984, 2nd edition. they write: The
letters X and Y in these expressions are variables; they stand for
(denote) unspecified, arbitrary sets.

Yes, pray reread. If you do not yet understand it reread again.

Therefore we can denote a set by X and we can say that the set X grows.

Wrong, again. Consider X a variable that stands for an unspecified
integral number. Can you state that the integral number X grows?
No. You can state that X grows, but not that the integral numbers
grow. In the above X and Y are variables that denote sets in every
instance. But X and Y are *not* sets.

That is nothing else than to say that the number of states in the EC
grows. Of course the number 6 has not gown to 25. But it is simply a
matter of definition, how one interprets "to grow" and "number".

In that case, please provide a definition.

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.

Wrong. Read my explanation above.

Wrong. Read my explanation above. I am talking mathematics.

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).

Hrbacek and Jech teach standard set theory including the fact that in
ZF everything is a set.

Yes? Do they define sets as allowed to grow? Not in the quote you supply.
There they talk about set valued variables that can grow.

> 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).

Hrbacek and Jech teach standard set theory including the fact that in
ZF everything is a set.

But I see nothing that states that a set can grow.
--
*** 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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