Fast answer on mathematics

I see that Wolfram now supports arithmetic with infinite cardinals!

Input “aleph 2 ^ aleph 2” and it answers “aleph 3 assuming the generalized continuum hypothesis.” “aleph 1 ^ aleph2” gives the same answer.

But input “aleph 2 ^ aleph 1” and the response is “no known simplified form.” Is that true? Or is the simplification just unknown to the Wolfram software?

Wolfram Alpha is wonky. The generalized continuum hypothesis says that “aleph” can be replaced with “beth”, and if you enter “beth 2 ^ beth 1” manually, you’ll see that it yields beth_2 again.

In general, in classical mathematics, (beth_a) ^ (beth_b) = beth_a if a > b, and beth_{b + 1} otherwise. (This follows straightforwardly from the definition of the beth numbers (in particular, that beth_{n + 1} = 2^{beth_n}) and the classical fact that the product of two infinite cardinals is just the maximum among them).

(Oh, I also assume the Axiom of Choice (which follows from the Generalized Continuum Hypothesis) in the previous post, if you care about such things)

(Further pedantry: But the use of the Axiom of Choice is only in the claim about products of infinite cardinals in general, and shouldn’t be necessary if we restrict ourselves only to considering the beth series or only to considering the aleph series.

Anyway. I’ll return to this thread later, perhaps.)

We’ve also had a (just barely) finite number of threads about this too, the most comprehensive apparently being this one which ran for about 15 years.

BTW, in case it matters to anyone, note that OP got bammed a few weeks ago.

14 years and half a month from first to last post, but the first 15 posts occurred within a 15.5 hour span in the year 2000, after which the thread went blank until being resurrected in August 2012 (not in response to any old post; just a new post by a new poster on the same topic, otherwise unrelated to everything previous), running from then until August 2014 for its remaining 2162 posts. For all intents and purposes, those were two separate threads: one which lasted less than a day, and one which lasted 2 years.

Also, there’s a jump from January 2013 to June 2014 in there. So, really, all together, about half a year of cumulative activity.

This video offers a beginner level explanation of different infinities.

That video starts off with one of my big bugaboos… “Infinity is not a number”, they say, and then they go on to illustrate all the ways in which infinity (or, rather, various different infinities) might well be construed as a cardinal number. Bleh.

Yeah, it bugs me, too. But I assumed that, since Dr. Grime has a PhD in mathematics, there must’ve been something I was missing.

Man, OP’s fate and the speed with which he met it makes me wonder how dangerous even thinking about infinity is.

Anything can be a number, depending on how you construct your mathematical rules. Under some sets of mathematical rules, infinity is not a number. Under others, it is. Under yet others, there is no single number called “infinity”, because there are multiple valid numbers which are infinite (maybe two, maybe an infinite number of them). And under yet other sets of rules, it not only isn’t a number; it isn’t even a recognized concept at all.