Re: Earth 8??

In article <LePOf.115941$QW2.61005@dukeread08>,
Ophidian <ophidian23@xxxxxxx> wrote:

Nathan Sanders wrote:

In article <suKOf.115920$QW2.35241@dukeread08>,
Ophidian <ophidian23@xxxxxxx> wrote:

Nathan Sanders wrote:

Give me a real number, I'll give you an integer.
Let's see who runs out first.

The usual proof goes as follows (we'll only consider positive reals
between 0 and 1 for clarity; that set is the same size as the set of
all reals):

List all of your numbers and my numbers, as follows:

1 0.3857354703483719453...
2 0.9264337854905487822...
3 0.2395549230438505875...
etc.

Now, it's clear that you've exhausted all of your numbers.

No, seriously.
How do I run out?

Because I said "list all of your numbers". Which positive integer do
you think is missing from the infinite list of positive integers that
begins with 1, 2, 3, and infinitely proceeds one-by-one? Prove that
your number is missing (I was able to prove it with mine).

The result is 0.439..., a number that is different from every other
number on the list, which means it *can't* be on the list yet, so it
must be a new number. You ran out, I didn't.

You just gave 1 more number, and a pattern for generating more.

Not just more... more that are *guaranteed* not to be on a countably
infinitely long list.

The reals simply cannot be put onto an ordered list. This isn't just
my idea; this dates back to Cantor, who originated the diagonal proof
I outlined above.

http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

I interate by 1 and keep pace fine.

No, because you've used up all the integers. Otherwise, it wasn't a
countably infinite list to begin with. That's the whole point; once
we've constructed an infinitely long list, all of your numbers are
there, but I can still prove that there are numbers not yet on my
side. It creates a contraction to assume that I can list all of mine,
because I can show that in fact, even with an infinitely long list,
there are still lots more numbers I haven't yet listed.

Note that Cantor's diagonal proof does *not* work for the even
numbers. If I create an infinitely long list of the even numbers next
to your numbers in pairs of (n,2n), then I construct a method for
generating an even number not on the list.

You're correct, it can't be done for infinite sets in general. It
*can* be done for *countably* infinite sets, however. the fact that
such a pairing exists is precisely why they are countable!

That is what I was trying to demonstrate.
1:1 alone is useless.
A determinitive, predictive rule applied against a known "countably
infinite" set is necessary to conclude a new set is also "countably
infinite".

A 1:1 mapping *IS* a "determinitive, predictive rule"! That's
essentially the definition.

"That's what a 1:1 correspondence means: that _every_ member of one set
can be paired up with a member of the other set with none left over."

That CAN be done with any two infinite sets.

No, it cannot.

If you have a countably infinite set and an uncountably infinite set,
they cannot be put into a one-to-one correspondence. *By definition*
of "countable".

You're conflating countably and uncountably infinite sets together.

As I said, pick one from mine, and one from yours.
There's a pair.
Continue until one set runs out.
It doesn't happen, ergo every element of one set CAN be paired to an
element of the other set.

Don't confound mathematical abstraction with physical reality. Of
course we would never run out in the real world, because human beings
don't live forever.

Infinity is an abstraction, and it has to be dealt with in abstract
terms.

B) A subset contains less elements than its parent set. (For
arguement's sake, ignore the special case of a set being its own subset.)

B is false; it's only true for finite sets.

Only if you assume all infinities are of equal size.

No. In fact, that's precisely the opposite of what I've said multiple
times. Countably infinite sets are all the same size. Uncountably
infinite sets are larger than that.

http://en.wikipedia.org/wiki/Countable_set

Two mathematically infinite sets have no requirement to be of equal
actual size, even if they are both "countably infinite".

If they are countably infinite, yes they do, by the very definition of
countably infinite.

I'm seeing a pattern here: your problem is not knowing the definitions
of the terms you're using. (Or perhaps creating your own definitions,
which you're free to do of course, but you need to realize that your
definitions are non-standard in mathematics.)

(Further, A is only
marginally true, because formally, set size is defined without
reference to numbers or elements, just equivalence in mappings
(one-to-one pairings).)

The size of non-infinite sets is independent of the number of elements
contained?

No... the notion of "set" and "size" is independent of the notion of
"number". Indeed, mathematicians take sets to be primitives, with
numbers derived from them. Thus, the sets {a,b,c} and {d,e,f} can be
shown to be the same size without doing any sort of counting or use of
numbers. We say they are the same size because there is a 1:1 mapping
between them: {(a,d), (b,e), (c,f)}.

A set has no size when not compared to another set?

Not really, no. Numbers are just special types of sets, so saying
that a set has "size 3" is just shorthand for saying that there is a
1:1 mapping between the set and a special set we have denoted with the
label "3" (often defined as the set { {}, {{}}, {{},{{}}} }.)

See http://en.wikipedia.org/wiki/Axiomatic_set_theory for more info.

If you really are interested in learning about infinity (rather than,
in typical Usenet fashion, just arguing for the sake of arguing and
stubborn unwillingness to admit that you don't know the material
you're talking about), then I suggest checking out the Wiki links I
gave you, as well as Googling for something like {countable infinite
"set theory"}, because Usenet is not an appropriate medium in which to
teach a course on set theory.

(And honestly, the web isn't either; you really should just take a
couple of college courses in set theory if you want to learnt his
stuff.)

Nathan
.

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