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).
(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.)
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.
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.
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.