## Tuesday, August 5, 2008

### Aleph-not is a large cardinal

The math level in this post may be a bit higher than the usual level in my math entries so feel free to skip it. I just have to share this because I found it absolutely amazing.

0 is a large cardinal. We just declare it to exist with an axiom that we all agree on. For all the common definitions of large cardinals 0 satisfies the requirements. To a lot of people who have studied model theory and set theory this will likely seem trivial, but this was pretty mind-blowing to me.

This information is courtesy of Harry Altman.

Etienne said...

It's completely unrelated to this post, but I just encountered a cute fallacious argument.

Suppose I show you two envelopes, one containing \$5, the other, \$10. I seal the envelopes and you choose one at random. I then offer to let you switch envelopes.

Let X denote the amount of money in your envelope. Then the expected value of switching is 1/2 * (2X) + 1/2 * (X/2) = 5/4 X > X, favoring switching, which is clearly absurd.

sniffnoy said...

Huh, do they not distinguish "ah" and "aw" up in Connecticut? I guess not. Because, um, it has an aleph number, just that aleph number is zero. Hence why it's called...

Joshua said...

Harry, I'm allowed to be a bit slow at times, ok? No need to rub it in so much...

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