Wikipedia:Reference desk/Archives/Mathematics/2011 September 16

= September 16 =

Aleph numbers
Hello, 2 questions. 1. The set (0,1) has a cardinality of $$\aleph_1$$, so can you say that if a point is randomly chosen from it is the possibility P(a) that it is a particular 0<a<1 $$P(a)=\frac{1}{\aleph_1}$$ (which I know is for all intents and purposes 0)? 2. How do you draw $$\aleph$$ with a pen and paper? All I can get is apparently a very uneven, unsymmetric, not aleph-y looking at all 'N'. 24.92.85.35 (talk) 21:51, 16 September 2011 (UTC)
 * First of all, it's not known whether the cardinality of the set (0,1) is in fact $$\aleph_1$$ or not. That claim is a statement of the continuum hypothesis, which can neither be proved nor refuted from the standard axioms of set theory, called ZFC.
 * It is an open question whether there might be axioms extending ZFC that would be sufficiently compelling to command a consensus among set theorists, that would either prove or refute the continuum hypothesis.
 * As for the probability question, if you choose a point from the uniform distribution on (0,1), the probability that it will equal some particular real number a between 0 and 1 is not merely "for all intents and purposes" zero. It's exactly zero.  A probability of exactly zero does not (contrary to a common misimpression) necessarily imply that the thing cannot happen &mdash; see almost surely for details. --Trovatore (talk) 21:57, 16 September 2011 (UTC)
 * Trovatore forgot to explain that it is not only exactly zero, but it is the real number zero as opposed to the rational number zero, which (according to Trovatore) is a completely different breed of cat. Bo Jacoby (talk) 09:37, 17 September 2011 (UTC).
 * No need to explain that. Clear from context. --Trovatore (talk) 09:42, 17 September 2011 (UTC)
 * In all fairness, in the thread you linked to, all-but-one editor supported Trovatore's point. — Fly by Night  ( talk )  13:14, 17 September 2011 (UTC)
 * Trovatore is right about this, and even if he wasn't, WP:POINT. -- Meni Rosenfeld (talk) 17:09, 17 September 2011 (UTC)
 * 1) The unit interval doesn't necessarily have cardinality $$\aleph_1$$; that's the continuum hypothesis. The probability of choosing a particular point isn't just effectively 0, it's exactly 0 (actually, this depends on the probability distribution used, but it's true for the standard one).  The expression $$\frac{1}{\aleph_1}$$ doesn't make sense, since division isn't defined for cardinals.
 * 2) I draw the diagonal first, then add the legs. They come out so-so.--Antendren (talk) 22:05, 16 September 2011 (UTC)

I draw the "legs" of the aleph as kind of integral-sign-shaped squiggles. That part of the question might be better for the language reference desk though. Never ask a mathematician how to draw a lower-case Greek "xi", either. Most just scribble. -GTBacchus(talk) 02:51, 17 September 2011 (UTC)
 * For drawing an Aleph, draw the Upper left lower right diagonal first. Now if you were to connect the other corners to the center, you would get an 'X' and if you were to connect them to the corners on the same side you would get an 'N', instead connect them to a point half way in between (the points 1/4 and 3/4 down that the primary diagonal. What also works is if the lines from "other corners" go to the halfway point vertically before heading to the quarter points). http://www.hebrew4christians.com/Grammar/Unit_One/Aleph-Bet/Aleph/aleph.html puts the joining points slightly closer to the center than to 3/4, but as long as the two that aren't the main diagonal meet at different places, you should be OK.Naraht (talk) 05:47, 18 September 2011 (UTC)

Just to be clear, since this is a point that some mathematicians not working in this area may be confused about: The cardinal number $$\aleph_1$$ is defined as the cardinality of the set of all countable ordinals. Whether it is the same as $$2^{\aleph_0}$$ is a perplexing question at best (and known to be independent of ZFC). Michael Hardy (talk) 17:27, 18 September 2011 (UTC)

As for how to draw aleph, I do the top-left to bottom-right diagonal and then add a couple of integral signs. Take a look at the picture I drew, using a fountain pen brush, in MS Paint. — Fly by Night  ( talk )  18:33, 18 September 2011 (UTC)
 * The legs should be closer to the center than what is in the picture (maybe about 35% and 70% from the bottom-right corner), and simpler - they should start at about 60 degree angle from the horizontal and end vertically, without squiggling. -- Meni Rosenfeld (talk) 04:57, 19 September 2011 (UTC)
 * Thanks for the advice, but I'm pretty sure that that's how I write it&thinsp;(!) If you write it in a different way, then draw a picture and share it; don't correct people. Different people have different hand-writing. Some people write curly xs like back-to-back cs while other write straight xs like it appears on your keyboard. To highlight the point: aleph is often written cursively like this . — Fly by Night  ( talk )  09:38, 19 September 2011 (UTC)

Why is division not defined for cardinals? --84.62.204.7 (talk) 15:33, 19 September 2011 (UTC)
 * Why should it be? — Fly by Night  ( talk )  16:15, 19 September 2011 (UTC)