# Fast answer on mathematics **URL:** \ **Category:** Factual Questions\ **Created:** [January 28, 2016, 12:57am UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438 "2016-01-28T00:57:06Z")\ **Posts on this page:** 12\ **Page:** 3
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [March 14, 2016, 8:45am UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/41 "2016-03-14T08:45:12Z")
> [@Asympotically\_fat](#): > > The cardinality of an infinite dimensional real vector space V will be the cardinality of the continuum if the dimension (which is itself the cardinality of the maximal linearly-independent subset) is less than or equal to the cardinality of the continuum. I see that Wolfram now supports [arithmetic with infinite cardinals](http://www.wolframalpha.com/input/?i=aleph+2+%5E+aleph+1)! 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? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 14, 2016, 7:40pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/42 "2016-03-14T19:40:25Z")
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). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 14, 2016, 7:46pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/43 "2016-03-14T19:46:39Z")
(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) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 14, 2016, 7:58pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/44 "2016-03-14T19:58:41Z")
(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.) ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [March 15, 2016, 6:45am UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/45 "2016-03-15T06:45:03Z")
> [@Anarcho\_Liberalist](#): > > Btw: Some time ago Cecil or the staff wrote a column about a related question - is .999~ the same as 1? [An infinite question: Why doesn’t .999~ = 1? - The Straight Dope](http://www.straightdope.com/columns/read/2459/an-infinite-question-why-doesnt-999-1) We’ve also had a (just barely) finite number of threads about this too, the most comprehensive apparently being [this one](http://boards.straightdope.com/sdmb/showthread.php?t=32760) which ran for about 15 years. BTW, in case it matters to anyone, note that OP got bammed a few weeks ago. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 15, 2016, 7:04am UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/46 "2016-03-15T07:04:38Z")
> [@Senegoid](#): > > We’ve also had a (just barely) finite number of threads about this too, the most comprehensive apparently being [this one](http://boards.straightdope.com/sdmb/showthread.php?t=32760) which ran for about 15 years. 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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 15, 2016, 7:09am UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/47 "2016-03-15T07:09:55Z")
Also, there’s a jump from January 2013 to June 2014 in there. So, really, all together, about half a year of cumulative activity. ---
**Author:** ![Chilliwack](https://avatars.discourse-cdn.com/v4/letter/c/e79b87/32.png) [@Chilliwack](https://boards.straightdope.com/u/Chilliwack)\ **Post date:** [March 15, 2016, 12:25pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/48 "2016-03-15T12:25:13Z")
[This video offers a beginner level explanation of different infinities.](https://youtu.be/elvOZm0d4H0) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [March 15, 2016, 5:12pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/49 "2016-03-15T17:12:13Z")
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. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [March 16, 2016, 4:41pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/50 "2016-03-16T16:41:11Z")
> [@Indistinguishable](#): > > 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. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [March 16, 2016, 5:19pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/51 "2016-03-16T17:19:12Z")
Man, OP’s fate and the speed with which he met it makes me wonder how dangerous even _thinking_ about infinity is. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 16, 2016, 9:05pm UTC](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438/52 "2016-03-16T21:05:04Z")
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. [Previous page](https://boards.straightdope.com/t/fast-answer-on-mathematics/744438.md?page=2)