Why is there no set of all cardinal numbers? If one accepts the axiom of choice and the continuum hypothesis, couldn't one construct a 1-1 correspondence between each transfinite cardinal number and its corresponding natural number? If you can construct this correspondence, then shouldn't you be able to then put them in a set?

Yes.

I have a set of all of them. I just choose not to show it.

I have a collector's edition of that set.

I have an autographed 1st edition of that set.


But seriously, why would that lead to a 1-1 correspondence? You would have all the natural numbers, but then the alephs.

hmmm, don't think i can afford to buy any cardinals.

Rockman, sometimes you scare me with all of your crazy math.
I feel like if I murdered someone you might be able to use your math and solve the murder, and then they would make a T.V. show about that sort of thing.

Or use it to conceive some kind of unholy doomsday device that could erase the life from an entire city. It would disable all electronic devices in the radius and the sky would cry snow like tears. The snow would literally cause your body to come apart and break down. The site would not be inhabitable again for many years.

Huh, after thinking over where I wrote "unholy doomsday device"...
Is there anything but unholy doomsday devices?

Would a holy doomsday device be an angel of the lord or mayhaps some rampaging elephant man? Maybe a simple man that wishes to have 72 virgins or perbe an enraged woman with 1000 arms?

Would a nuclear warhead blessed by The Pope that used hydrogen stripped from holy water be considered a holy doomsday device?

Ya know, with all these 'imaginary' numbers and 'cardinal' numbers, mathematics is starting to look suspiciously like a religion.

Yes. I have evidence that proves this theory. I just choose not to show it.

The type of mathematics I am interested in does not have any real world applications.



Cardinal numbers are simply numbers which describe the size of a set. Imaginary numbers was a play on words by mathematicians (they have a sense of humor). The set of real numbers was supposed to include the totality of numbers when it was created, but then someone realized that the square root of negative one was not included in this set, and rather than just saying it doesn't exist, you can give it a name, and see what happens from there.

I've gotten a partial explanation so far on my original question.

Apparently there are more cardinal numbers than I had considered, and for many of them, their constructability is in doubt. The inaccessible Cardinals, for instance, are quite difficult to .... access :P

http://en.wikipedia.org/wiki/Inaccessible_cardinal
If you like math without any numbers :)

I have to ask, what is your fascination with this kind of math?
What do you use it for, or do you simply study it?

I find it fun to think about. And I am considering going back to school to get a PHD in mathematics and become a math professor.

I enjoy logic :)

I think you should do that, yes. If you understand fully that wiki link you must have a gift.

It makes me feel undereducated, well in math anyhow.