Now for something completely different.

[Ham]

First: some vocabulary..

Cardinality: the number of elements in a set

A Set: A collection of objects. They can contain any members, usually numbers, for mathematicians.

sets can have a finite number of elements, or infinite..

If sets have the same quantity of numbers, they have the same cardinality, by definition. (Maybe!) more to come..

I wish Walter would come here. I know he's seen all of this at least once..

[Ham]

So one is confronted with different sets of numbers. There are the Natural numbers.. they are called the counting numbers. {1, 2, 3, 4, 5, ...} and then there are other sets of numbers: the Integers. 0, 1, -1, 2, -2, 3, -3...

which set has more numbers, or members? That is the first big question..

[Ham]

Maybe a hint. Any set one can list in a Roster is countable. That is, there is the first element, the second, the third, etc.

Don't say this is too hard. This is 1800's technology. We are in what now.. 213 give or take a few years past all of that..

[Ham]

So nobody wants to attempt to answer the first question. For shame, for shame..

[Ham]

This is where Cantor was brilliant. Of course, Kronecker persecuted Cantor, not to Cantor's grave, but to Kroneckers. Kronecker described one of the greatest minds of the 19th century as a mathematical charlatan, a corrupter of the youth..

Kronecker was a dull little bastard. Well, he showed a little brilliance, from time to time..

[Ham]

So, what if we just re-named numbers, and just gave them a different label.

Instead of 1, we call it 1 squared, which is 1.

Instead of 2, we call it 2 squared, which is 4.

Instead of 3, we call it 3 squared, which is 9.

Here is where the problem (question number one) exists.

We've skipped over a lot of numbers so far.. 2, 3, 5, 6, 7, 8.. but these sets STILL have the same number of members.

[Human without the bean]

So nobody wants to attempt to answer the first question. For shame, for shame..

A little late. Numbers are infinite. You know, like Buzz Lightyear would say," to infinity and beyond".

Edited May 2, 2013 by Human without the bean

[Ham]

Hey.. if you want something completely different.

May I suggest:

Yes, I've listened to this on multiple occasions.

[waysider]

[Ham]

We've already lost a hand full of numbers, and in the final tally it has made no difference..

[Ham]

This one also has a couple of numbers.. 1967, and 19..

[waysider]

[Ham]

This is among some of the great secrets of the Universe..

Man, reluctantly raises his head above the mire of society, and sees, and maybe even understands the nature of the first infinity..

hey.. the Greeks didn't dare go there. Archimedes probably knew more, but wasn't inclined to say..

now we are talking technology long before the 1800's..

[Ham]

Some Religious Folk thought Cantor's work to be Blasphemy.. but it was the wrong era. They didn't exactly succeed in raising religous persecution and such. Cantor held the belief that God revealed to him the Transfinite Numbers.

what are Transfinite numbers anyway.. anybody want to know?

This was a development in the 1800's. I mean.. come on. Great great gramps can't be so smart that we can't understand him, can he?

The actual mechanics behind the theory is really pretty easy to agree on..

if one can find a one to one correspondence between sets, no matter how different they might be.. one has then shown than one set is equivalent to the other..

if you want the english version, I will give it.

[Ham]

the problem with Kronecker's public railing against the Transfinite Numbers. They were not exactly numbers, to begin with They were more descriptions of the properties of subsets of numbers.. the first one is Aleph null. That one describes the set of Natural Numbers, or the counting numbers..

Its a frigging description. Not an exact quantity..

[Ham]

As far a history is concerned. Kronecker eventually died. And the Mathematical World embraced Cantor..

[Ham]

Kronecker had his little Delta. Then passed off, quietly into obscurity..

Argo has to permit infinite passage. Always room for one more..

[Human without the bean]

The actual mechanics behind the theory is really pretty easy to agree on..

if one can find a one to one correspondence between sets, no matter how different they might be.. one has then shown than one set is equivalent to the other..

if you want the english version, I will give it.

I thought that this was english. What was that song Heart did , oh yeah, the dog and the butterfly. Let's say, the dog follows the butterfly around eating nectar from a flower and then, the dog eats the flower.

By that definition are they equal or not, I'm wondering?

[Ham]

Perhaps. But then again.. one partakes of the flower, the other consumes it..

[Human without the bean]

.....and if the dog comsumed the butterfly, which is more likely to happen, then we have nothing.

Is nothing a number?

[Ham]

Technically.. the empty set is both closed and open.. I don't know if that helps here..

[waysider]

.....and if the dog comsumed the butterfly, which is more likely to happen, then we have nothing.

Is nothing a number?

[Twinky]

Are you a little bit bored with what's going on in your life right now, Squirrel? :rolleyes:/>

[Ham]

not quite..

On a graduate level I struggled with Real analysis..

now that I have a few minutes to breath.. I find that it is actually interesting.. the real number system.. is unbelievably comprehensible.

I'm just looking at it in terms of the tools we use in a basic algebra class..

Now that I can actually enjoy it for what it is..

a convergent series, but not absolutely. One can with careful re-arrangement, approximate any real number. That is, if I understand this correctly.

In other words. It looks simple.. 1 -1/2 + 1/3 - 1/4 + 1/5 - 1/6 ...

this sums to a real number. If one chooses a different arrangement of the numbers.. one can produce any real number. That it is possible is one thing. To choose the correct re-arrangement is yet another.

Weierstrass figured this one out. Not how to correctly choose, but that it is possible to so choose..

From a fundamental algebra point of view, yes, we know how to add fractions.

Am I bored? No.

[Ham]

Don't say its too hard or incomprehensible. Weierstrass lived in the 1800's.

He ran with a "bad crowd" though.. among whom included Dedekind, and the Heretic Himself, the Inventor of the Evil Numbers.. Cantor.