Stuart Explains Set Theory (DB)

ASCIIN SERIES
== Introduction ==
When Asciin was ascending upward through the layers of the cosmology, he had to traverse numbers larger than the infinite. Much of the explanation for this real field of study was omitted due to being uninteresting to the layman, but for those who want all the juicy details, this page exists.
== Aleph-1 ==
"Start by thinking of infinity," Stuart instructed. "Think of a list, with infinitely many items on it. The list is numbered. What happens when you add another? Infinity and all of the finite numbers have already been used."
"Mhm."
"So, it has to be infinity plus one. Then infinity plus two, three, four, infinity, infinity plus infinity plus infinity infinitely... And then infinity times infinity times infinity an infinite number of times. This isn't actually a bigger infinity, but it's the process we use to reach the first one. Now, let's denote this infinity to the infinitieth power to the infinitieth power, infinitely repeating, as Epsilon-zero. Then we have Epsilon-zero plus one, two, three, infinity, itself, times itself, to the power of itself, to the power of itself to the power of itself repeating itself many times; this is Epsilon-one. Imagine this same process, reaching Epsilon-infinity, Epsilon-Epsilon-zero, Epsilon-Epsilon-Epsilon-zero, Epsilon-Epsilon-Epsilon-Epsilon-Epsilon-etc, repeating until there are as many epsilons in the number as there are epsilon numbers, ending the hierarchy in a peak called Zeta-zero."
"And from there it just repeats itself forever," Asciin finished.
"Exactly!" Stuart praised him. "Now, the first genuinely greater infinity is called Aleph-one. It's basically the set of all numbers on the list. including everything from the hierarchy we just talked about. Think of it as not the final point on the list, but subsuming the whole list, because you can always add one more after the end of the list, and Aleph-one contains that too."
== Further Alephs ==
"What's the next one, Stuart?"
"It's the same principle, really. Everything here will be repeated until we get to large cardinals. So, imagine a list where the final number is Aleph-one. Aleph-two is the set of all extensions of that list, and entirely beyond it. Keep in mind that Aleph-one times Aleph-one already vastly eclipses anything from the previous hierarchy in its proportional leap."
"And I imagine it's the same from there," Asciin thought aloud, "with Aleph-three, four, five, infinity, Aleph-Aleph-one, et cetera, with each subsuming every extension of the list analogy from the previous one."
"Yup!"
== Inaccessible and Mahlo Cardinals ==
"Next is the inaccessible cardinal, which will probably be the last one I can adequately explain to you. Basically, it can't be reached by the Aleph hierarchies, hence 'inaccessible'. Easiest way to imagine this is to think of the difference between infinity and zero, because that's basically the difference between an inaccessible cardinal and infinity." Asciin imagined that, and...
"Holy- I just grew more in that one jump than I have since I started!"
"Told you it was big. Next, imagine something inaccessible to the inaccessible. Take the whole repetition of the Aleph hierarchy, apply it to the inaccessible cardinal, and imagine what could be inaccessible to that. Don't just imagine infinity to zero again; if one is zero to an inaccessible cardinal, an inaccessible cardinal is zero to an inaccessible cardinal multiplied by itself, which isn't even the inaccessible equivalent of Epsilon-zero. Imagine something completely outside of any and all hierarchies of inaccessible amounts, and you've reached a one-inaccessible cardinal. Then a two-inaccessible cardinal is inaccessible to one-inaccessible, and three-inaccessible is..." As Stuart prattled on, Asciin continued to climb, but he could still feel that the one he was after was far above him. He skipped ahead to what he mentally dubbed the "final level of the inaccessible hierarchy, which there can be no further inaccessibility from". Upon relaying this to Stuart, he was informed that this was called a hyper-Mahlo cardinal.
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Category:Canon
Category:Divergent Branch