"The universe can and will create itself from nothing."

[Jay313 (Jay Johnson)]

@Mervin_Bitikofer

I tell my calculus students that they can give more specific and informative answers.

I keep forgetting you’re a math guy, as well as @mitchellmckain.

@heymike3

From the little I’ve gathered, aleph-O represents the natural numbers, and could be pictured as (X, X+1, X+2…) This can also represent the coordinate system for a multi-dimensional space. Aleph-1 can then represent the reals between 0 and 0.00000001(X<—>nonX) and still be considered larger than aleph-0. Aleph-2 seemed like a kind of “natural number” as it is the result of function that is applied to aleph-1. This comparison to natural numbers could be totally wrong. But it was confirmed for me by someone who was knowledgeable in the subject. And the comparison explains for me why it is said there are an infinite number of aleph values.

@mitchellmckain

When I first heard about them the examples were…

aleph 0: the natural numbers
aleph 1: the real numbers
aleph 2: the number of possible functions on a real interval.

I thought this article was fascinating and it seems to answer the questions posed above. I’ll leave that judgment to the more mathematically educated, so please weigh in. If you have any interest in the history of mathematics, the article is definitely a must read. For the rest of us, I’ll post some TL/DR screenshots as a half-baked summary.

(Okay, in hindsight I went nuts on the screenshots, but the article’s a freebie so cut me some slack.)

https://www.cantorsparadise.com/the-nature-of-infinity-and-beyond-a05c146df02c

One for @Mervin_Bitikofer

Edit: Missed this one. Sorry, Merv

The conclusion seems to refute this quote that Mike found:

@heymike3

Aleph 1 is just the next largest infinity after the integers, which is not necessarily the reals. The statement that aleph 1 is the reals is the continuum hypothesis which can neither be proven nor disproven under the usual axioms of set theory.