Re: Ultimate debunking of Cantor's Theory

In article <1184388829.950882.95400@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Calvin <crice5@xxxxxxxxxxxxxx> wrote:

On Jul 14, 12:37 am, Rotwang <sg...@xxxxxxxxxxxxx> wrote:
Calvin wrote:
On Jul 14, 12:11 am, Proginoskes <CCHeck...@xxxxxxxxx> wrote:
Nope. This is the mistake: You are assuming that N can
be constructed in the first place.

I dealt with that in my proof, below, part of which is

"...by the definition of N, which is that
it is comprised of the elements of the sequence 1, 2, 3, ...
(or if you prefer, 1, (1+1), (1+1+1), ...)"

That clearly is a valid construction, though it
does not use set theory notation, which no doubt
would make the definition more elegant.

You are still missing the point, which is that simply specifying what
you want to be in your set does not mean that such a set exists. You
acknowledged before that Russell's paradox proves that there is no set
of all sets. But couldn't an argument similar to the one you use
above, if it were correct, show that there is such a set? "...by the
definition of X, which is that comprised of all x such that x is a
set".

No, "set of all sets" is a collection of words, like
"set of all elephants ice skating". Just because you
can say words that convey a thought about a set, does
not mean that such a set exists.

But the set of natural numbers does exist. It can be
constructed, demonstrated, and specified down to the
last detail. Ask me what the nth element is, and I
can tell you its value (n). It's hard for me to imagine
anything in mathematics that is more demonstrable than
the set of natural numbers.

While I can see numerals (names of numbers) I have never seen a number.
While I have seen the names of sets, I have never seen a set itself.

It is only our familiarity with numbers and their properties that allows
us to conflate those names with the named in so many instances without
causing problems.

The fact is that you cannot simply declare that any collection of
elements is a set;

That is a true statement, but that is not what I did.


this is what Franz meant when he wrote that

"What Russell's Paradox actually DOES show is that a set theory with
"unrestricted comprehension" is inconsistent (i.e. is not viable)."

The axiom of infinity states that the set you described exists (or
more precisely that a set containing all the elements you specified
and possibly some other elements exists, from which one can construct |
N). I do not believe that it is possible to prove that such a set
exists without the axiom of infinity.


That's really your motivation isn't it?
How dare I claim and prove that an infinite set
exists, without invoking the holy 'axiom of infinity."
.

Relevant Pages

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