Re: abundance of irrationals!)


aeo6 Tony Orlow wrote:
> imaginatorium@xxxxxxxxxxxxx said:
> >
> > aeo6 Tony Orlow wrote:
> > > mueckenh@xxxxxxxxxxxxxxxxx said:
> > > >
> > > >
> > > > Remember: Every set of even numbers contains numbers which are
> > larger
> > > > tan the cardinality of that set. This is valid for *every* set
of
> > > > finite numbers. There is no reasonable arguing claiming that
this
> > > > situation should change in case of an "infinite" set.
> >
> > Yes there is - it's the error contained in the "salient point"
below.
> >
> > Also some
> > > > elements of an infinite set of even numbers would necessarily
> > surpass
> > > > the cardinality.
> > > Yes. The **salient point** is that the set size can never be
LARGER than
> > the
> > > largest element.
> >
> > Absolutely! The set size can never be LARGER than the largest
element.
> > Shout it from the rooftops! If you can find the largest element,
the
> > set size cannot be larger. Great! Do we all agree now??
> >
> > Uh-huh. Problem. What happens when there is NO largest element?
Then
> > the set size cannot be larger than, er, um, well, come in Houston,
we
> > don't seem to have a largest element here.
> >
> > Brian Chandler
> > http://imaginatorium.org
> >
> >
> Brian I already responded to that, you fool.

Insults are very unproductive. You do not help by throwing words like
"moron" at people like Randy Poe, who show unimaginable patience in
pointing out your mistakes over and over again. Anyway...

> You claim an infinite set
> containing only finite values. You claim the argument fails because
there is no
> largest element, but whatever it is, it's finite. Well, if all values
in the
> set are finite, then they are all less than infinite, **so** the set
size cannot be
> infinite. QED

Why "so" (my asterisked emphasis)? The set consists entirely of finite
values. The fact that there is an unlimited number of them does not *of
itself* imply that any of these values is itself finite - as you
yourself agree, in the case of (e.g.) the rationals in [0, 1]. So why
do you think it does in this case? Well, look at _your own_ argument.
The "salient point" you say is something to do with the largest
element; but then when we point out that no largest element exists,
somehow this "salient point" doesn't matter any more?

Anyway, it's not surprising you think there's something wrong with set
theory, given your extraordinary misunderstanding of it. One of the
most basic things set theory allows us to do is to consider the element
of all elements (within some universe of discourse) having a particular
property. When we say "The set of all x having property P", we mean
(obviously) that every single element x of this set does have the
property P. Is this clear to you?
If so, when I say "The set of all integers x, each x being a finite
integer", why do you claim that by some mysterious process at least one
other element y, not having the property of y being a finite integer
will somehow creep in?

Several people have asked you - over and over again - well, consider
the set with all the 'naughty' elements removed. In terms of your
"integers" (actually the 10-adics) including "numbers" with an
unlimited number of nonzero digits, do you think that if you consider
any one of these it must either have or not have an unlimited number of
nonzero digits? If so, how many of then do *not* have an unlimited
number of nonzero digits?

> You certainly draw a lot of conclusions about a set which you cannot
enumerate.

Which set (precisely) is that?

Brian Chandler
http://imaginatorium.org

.