Re: Ultimate debunking of Cantor's Theory

Calvin wrote:

"...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.

Exactly.

But the set of natural numbers does exist. It can be
constructed,

How? You wrote:

"...by the definition of N, which is that it is comprised of the
elements of the sequence 1, 2, 3, ..."

You merely specified what elements you wanted your set to consist of,
just like I did. You didn't prove that such a set exists, and my
example was intended to illustrate that merely specifying elements
does not constitute a proof of existence. Perhaps you are applying
some general principle whose hypotheses are met by your construction
but not mine. If so it is up to you to say what that principle is, and
prove that it holds. Perhaps, for example, you think that specifying
elements in the form of a finite list followed by an ellipsis will
always yield a set.

Ask me what the nth element is, and I
can tell you its value (n).

Sets don't have "nth elements". They are unordered.

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.

So what did you do? What axiom or theorem were you using to deduce
that your ability to write "by the definition of N, which is that it
is comprised of the elements of the sequence 1, 2, 3, ..." means that
N exists?

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."

For ***'s sake. It painfully clear that you don't know the first
thing about ZF set theory or how axioms are used to prove theorems.
Since you seemed sincere I was trying to help you, but you responded
by questioning my motivation. Piss off.

.

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