Talk:Aleph number/Archive 1

Algebraic infinity
I'm not sure about the paragraph discussing the aleph numbers and algebraic infinity - I think it might cause some confusion, since it states that the two are unrelated, yet aleph numbers were developed to deal with infinite sets. Perhaps something that explains the difference? -- ocmpoma
 * I agree, I'm removing/altering the line that says infinity is just infinity, due to the fact that there ARE infinities of different size. This is proven by the fact that infinity-infinity != 0. Shadowblade (talk) 05:42, 25 August 2008 (UTC)


 * Actually I thought that line was helpful. While there are different kinds of infinite numbers, the calculus $$\infty$$ really is just $$\infty.$$
 * CRGreathouse (t | c) 13:46, 26 August 2008 (UTC)
 * As pointed out at ordinal arithmetic, "Ordinal addition is left-cancellative: if α + β = α + γ, then β = γ." It follows that (α + β) &minus; α can be defined unambiguously as β.   (It does not however follow that (β + α) &minus; α = β because of the ambiguity: since 1 + ω = 2 + ω, what should be the result of subtracting ω?  1 or 2?)  With this definition of subtraction, and noting the identity α + 0 = α, we have α &minus; α = (α + 0) &minus; α = 0 for any ordinal α, whether finite or infinite.  With the definition of a cardinal as a particular kind of ordinal, this also holds for all cardinals and therefore for all aleph numbers.  Whatever conventions are observed in calculus are irrelevant to this argument.  --Vaughan Pratt (talk) 22:17, 11 June 2009 (UTC)

Merges
I've merged together the aleph-null, aleph-one articles, and the fragement from the aleph article. I think this is justified because the alephs, particularly aleph-null and aleph-one, are so closely tied together that they should be discussed in the same article. This now needs copyediting and review by someone else. -- The Anome 11:11, 11 Aug 2003 (UTC)


 * good idea. It looks pretty good to me! :) -- Tarquin 13:52, 11 Aug 2003 (UTC)


 * Looks good to me, I'm in disagreement with Michael Hardy regarding the definition of Aleph-1 and haven't had the time to actually follow through on some of the research. I claim that that Aleph-1 (with the Axiom of Choice) = the smallest infinite cardinality greater than Aleph-0, but not necessarily equal to the continuum, while he claims that Aleph-1 = the continuum, but it is uncertain whether there are values between the two.  (Hope I summerized the disagreement properly).  If there are other people with knowledge here, I'd appreciate their input. GulDan 17:48, 11 Aug 2003 (UTC)


 * You have not summarized the disagreement at all correctly. Michael Hardy was pointing out that "aleph_1 is the next cardinal after aleph_0" is a theorem of ZFC, not the definition of aleph_1. He certainly did not say that aleph_1 = the continuum. --Zundark 21:10, 3 Sep 2003 (UTC)


 * Thanks for the correction. I realized later that my def was off too, but he was still accepting the CH which was my primary complaint.  However, all is fixed now.  Once again, thanks for the clarification. GulDan

Countable subsets
I have read in quite a few books on the so-called "extremely useful" and "remarkable" property of Aleph-one that every countable subset of it has an upper bound in it. But the only example that I know that actually uses this fact is trying to find an explicit description of the sigma algebra generated by a collection of subsets. I would guess that it is useful for closure with respect to countable operations (such as countable unions + complements in the sigma algebra case). But if anyone has more examples or can elucidate this issue... suggestions welcome (and add it to aleph-one).

Choni 10:10, 21 Nov 2003 (UTC)

Consistent glyph
Is there more work needed to get total consistency in the typo-style for the aleph-null glyph? - Bevo 20:04, 4 Mar 2004 (UTC)

Pronunciation
Should we mention that Aleph-null is also pronounced Aleph-nought? I had never heard -null before. —Preceding unsigned comment added by 161.32.76.138 (talk • contribs)
 * Really, I've only ever heard null, never nought. Perhaps it's a Brit English/American English thing?  Americans generally never use nought for anything.  Ford MF (talk) 18:02, 31 July 2008 (UTC)
 * In my experience aleph-naught is the most common pronunciation among mathematicians, on both sides of the pond. Usually when you hear aleph-null it's from one of the popularizers. Not quite sure how that happened. But it's not limited to the alephs; in math and physics naught is a common way of saying subscript zero in general. --Trovatore (talk) 02:18, 1 August 2008 (UTC)
 * I've only ever heard naught, and I live in the US. Mentioning..... Shadowblade (talk) 05:46, 25 August 2008 (UTC)

I have only heard Aleph null. I live in the US. We use "nought" or "naught" when reading subscript 0 for other variables, but I think people have always said "null" for Aleph. I ran a Google Fight and "Aleph null" had 16500 hits, "Aleph nought" 1990 hits and "Aleph naught" 4210 hits. So Aleph null would seem to be the prefered pronunciation, at least on the Internet.--seberle (talk) 15:49, 13 February 2009 (UTC)
 * A lot of those hits are probably from popularizers, who for some reason have long preferred null. Among the people who actually work with it (meaning set theorists) it's mostly naught. --Trovatore (talk) 20:15, 13 February 2009 (UTC)
 * Actually the most common term on the Internet by far seems to be "Aleph zero" (66,600 hits). I have in fact seen this usage from some respected authors. --seberle (talk) 23:33, 21 February 2009 (UTC)

Whoops. I forgot to include quotation marks when googling. The actual number of hits is: But you are right, we should find out what term set theorists actually prefer. --seberle (talk) 23:40, 21 February 2009 (UTC)
 * "Aleph naught" 497 hits
 * "Aleph nought" 607 hits
 * "Aleph null" 6300 hits
 * "Aleph zero" 9670 hits

Ok, I came across the following footnote in a recent paper by Sanchez (http://arxiv.org/PS_cache/arxiv/pdf/0804/0804.2934v1.pdf):
 * Although the usual way of reading $$\aleph_0$$ is aleph-null -it can also be read as aleph-zero- the original english translation by P. E. B. Jourdain of Cantor’s Beiträge was ”aleph-zero”. Section 6 is entitled ”The Smallest Transfinite Cardinal Number Aleph-Zero. The current English edition of Cantor’s Beiträge is from 1955.

--seberle (talk) 00:03, 22 February 2009 (UTC)

Aleph two
I think the aleph two section is completely wrong; I'm going to remove it completely. Aleph two is not as interesting as some other alephs anyway, like aleph omega-sub-six (which cannot be equal to the continuum -- I think). —Preceding unsigned comment added by 137.131.236.80 (talk • contribs)


 * It was wrong insofar as it said "second smallest" instead of "third smallest infinite". I'm not sure it was best to remove it completely, rather than just correcting it.


 * Can you explain why aleph_{omega_6} is not the continuum? I thought it was known that this could not be proved in ZFC (assuming ZFC is consistent), because aleph_{omega_6} has uncountable cofinality. --Zundark, 26 Jul 2004


 * Ok first off, what is aleph_{omega_6}? Why omega_6? Just plain aleph_omega, the first aleph after the alephs with finite indices, has countable cofinality and hence cannot be the continuum.-- Choni

Aleph-not?
I've heard it referred to as Aleph-not on multiple occasions, perhaps a page for Aleph-not should be redirected to here? —Preceding unsigned comment added by 24.214.14.35 (talk • contribs)

actually its spelled "Naught" —Preceding unsigned comment added by 69.140.206.114 (talk • contribs)

See the above discussion under "Pronunciation." --seberle (talk) 23:52, 21 February 2009 (UTC)

Possible origins
This section is mere speculation at this time. Should be sourced or deleted. BTW it seems a little implausible to me that Cantor used "aleph" to suggest an identity between mathematical infinite and theological infinite. I think that would have struck him as sacrilege; as I understand it he chose the term "transfinite" as opposed to "infinite", at least in part, specifically to avoid such an implication. --Trovatore 19:46, 24 July 2005 (UTC)

∞ isn't just ∞
Maybe i'm wrong but i think that ∞ isn't just ∞.. There are different orders of infinity.. e.g.: Limit(x / e^x, x->∞) -> 0, so e^∞ is an infinite grater than the mere ∞. —Preceding unsigned comment added by 213.156.52.121 (talk • contribs)


 * You're probably thinking of Big O notation or asymptotic analysis, which don't have much to do with infinite cardinals. Note that $$\lim_{x \to \infty} f(x) = \infty$$ simply means that f(x) increases without bound, that is, f(x) can be made larger than any specified real number by choosing an appropriate value for x. —Keenan Pepper 15:28, 31 January 2006 (UTC)

No, there is a sense that there are different orders of "plain" ∞, specifically, infinite sums like
 * $$\sum_{i=0}^{\infty} f(i)$$

add together a countable number of terms, whereas integrals like
 * $$\int_{a}^{\infty} f(x)\,dx$$

are sums of an uncountable number of values. — Loadmaster 17:46, 2 October 2007 (UTC)

Internet Explorer
I notice JRSpriggs always undoes changes that make use of the ℵ symbol and prefers that a image is generated for it instead because Internet Explorer doesn't support them. I just checked here and I can see them just fine in Internet Explorer 6. I doubt it's an issue with IE, more likely is that JRSpriggs has font problems. From my reading of the Wikipedia rules it appears that it is more accessible and is better to use the actual characters in cases such as this? —The preceding unsigned comment was added by 71.193.190.213 (talk • contribs).


 * I'm not sure what JRSpriggs's technical problems are, but he also can't view the "element of symbol", and so won't allow them in articles he works on either. It's a sad state of affairs.  -lethe talk [ +] 04:33, 11 June 2006 (UTC)


 * I agree, there's nothing wrong with Internet Explorer. (Wait, wtf did I just say? I meant it's not IE's fault in this particular case...) JRSpriggs just doesn't have a font that has the aleph character. —Keenan Pepper 16:54, 11 June 2006 (UTC)
 * That's what I initially thought as well, but when this first came up, I think we established that this is not the problem; the character displays correctly for JRSpriggs in Firefox, but not in IE. So his system must have a font with the symbol, right?  If IE works for some people and and JRSpriggs does have the fonts necessary, I have no idea why his system won't display the symbols, but apparently he's not the only one (his nephew as well). -lethe talk [ +] 17:49, 11 June 2006 (UTC)


 * Here's some history of this issue. This was first brought up here, and afterwards JRSpriggs changed from unicode to inline PNGs in about a dozen different articles.  I then brought it up at the project talk page.  -lethe talk [ +] 18:11, 11 June 2006 (UTC)


 * Here's how I see it:
 * JRSpriggs has the right font.
 * Others, using IE and the right font, can see &alefsym;, but JRSpriggs can't.
 * So JRSpriggs' copy of IE must not know how to find the right font, or hasn't been told that it's supposed to use it.
 * This is a guess, of course, but it seems like the place to look. JR, could you go into Preferences in your copy of Firefox and figure out where it's looking for fonts, and then tell IE where that is, or else copy them into your C:\Windows\Fonts directory and "install" them or whatever it is you have to do? --Trovatore 19:44, 11 June 2006 (UTC)


 * So actually, if you follow the link above to the discussion on the math project talk page, you will find that you also verified that your own copy of IE exhibits the same behavior. Maybe tomorrow I will go find a Windows computer in the lab and check it myself. -lethe talk [ +] 20:24, 11 June 2006 (UTC)

Both my copy of Firefox and my copy of Internet Explorer say that they are using Unicode (UTF-8) as their character encoding. Auto-select is turned off for both. Are those the correct choices? JRSpriggs 03:11, 12 June 2006 (UTC)

bijective/bijection
Regarding the recent, shall we say, edit skirmish, here's how I see it: JR wants "bijective" and "one-to-one" both to modify "correspondence". That's perfectly cromulent grammar, but you need to punctuate it as follows:
 * ..a bijective, or "one-to-one", correspondence...

because otherwise it's not clear that "correspondence" is supposed to distribute over the "or". If you do it like that, though, it becomes problematic to wikilink, at least if you want to wikilink both bijective and one-to-one correspondence.

In fact, though, bijective and one-to-one correspondence are both redirects to bijection, so you really shouldn't wikilink them, per the MoS.

So no vote, per se, on my part as to the conflict between the two editors; just a suggestion that if JR's sentence structure is used, then the commas should be placed as I have recommended, and "one-to-one correspondence" should not be wikilinked. --Trovatore 16:45, 26 August 2006 (UTC)

Definition
I'm a little concerned that the article "Aleph number" doesn't actually define $$\aleph_n$$ in generality. It uses the ZFC definition of the cardinal infinite number just larger than $$\aleph_{n-1}$$, but without the axiom of choice this isn't unique.

CRGreathouse (t | c) 04:10, 3 January 2007 (UTC)

Notice ω1 is an uncountable set
We currently say this:

Can we explain this a bit better? How is the reader supposed to "notice" that it's uncountable? It's not stated in the ordinal number article. --Doradus 15:52, 13 June 2007 (UTC)


 * "Notice" is math jargon for "it can be proved that". The ordinal number article says "Also, ω1 is the smallest uncountable ordinal".  I rephrased that sentence, but feel free to edit it more thoroughly to explain. &mdash; Carl (CBM · talk) 14:05, 14 June 2007 (UTC)


 * I have tried to explain it.--Patrick 13:49, 15 June 2007 (UTC)


 * I added the section Ordinal_number.--Patrick 14:13, 15 June 2007 (UTC)

Choice
Do you have to assume choice to show aleph null is the smallest infinite cardinality? This would surprise me if true... 137.222.230.13 (talk) 22:02, 28 February 2008 (UTC)
 * Well, you don't need any choice to show that there's no smaller infinite cardinality. You do need a tiny bit to show that there's no incomparable cardinality. --Trovatore (talk) 22:51, 28 February 2008 (UTC)
 * Should I modify that part to assume only Countable Choice and/or mention that choice is not needed to prove there are no smaller cardinals? JumpDiscont (talk) 22:53, 28 September 2009 (UTC)

Missing ℵ
This article should mention the Unicode character you can get by using the symbol character &amp;alefsym; (ℵ) rather than the TeX math markup aleph character $$\aleph$$ ($$\aleph$$) used in this article. That character redirects here, and it is used, for example, in the real number article, which is the only reason I know about it. Can someone fill us in on its Unicode number and the like, and that of the different Unicode aleph character which appears in the Hebrew alphabet (א} (which redirects to the aleph article, unlike the &amp;alefsym;), etc.
 * $$\aleph \,$$ math markup
 * &alefsym; Unicode alef symbol
 * א Unicode Hebrew letter aleph

Gene Nygaard (talk) 11:57, 8 April 2008 (UTC)
 * Does that one still cause problems in IE? It would be nice to use it instead of PNGs if it works for Microsoft users. --Trovatore (talk) 16:52, 8 April 2008 (UTC)
 * I opened up IE 7 and took a screenshot: [[Image:Aleph ie.png]]
 * It looks much better in Firefox, but all are serviceable even in Internet Explorer.
 * CRGreathouse (t | c) 19:07, 8 April 2008 (UTC)
 * I'll have to take that back halfway. The unicode aleph symbol doesn't show up for me at all in IE 6 (different PC). The aleph letter shows up fine. CRGreathouse (t | c) 23:08, 8 April 2008 (UTC)


 * I can't see the unicode character either; nor can I see the element symbol above. In fact, I've had trouble seeing a lot of unicode on Wikipedia.  My IE7 seems fine in all other ways.  I'm not sure why I'm having this problem.Eebster the Great (talk) 00:05, 9 April 2008 (UTC)


 * Alefsym and the unicode alef are still not readable in my copy of Internet Explorer. However, I only use Firefox now when reading Wikipedia, and I have no trouble in Firefox. JRSpriggs (talk) 08:49, 9 April 2008 (UTC)

too technical
I added the Template:Technical header to the article because I think it's mostly impenetrable to nonmathematicians. I'm a reasonably intelligent and well educated adult, and halfway through the lede, I was completely out to sea. As a general encyclopedia, we should endeavor to allow our readers to at least grasp the concepts we are talking about, and I think we can do it in a way that does not sacrifice technical accuracy or comprehensiveness. But after reading this article, I am no closer to having any idea what an aleph number actually is than I was half an hour ago before I read it. I understand that this is a complicated subject, but at the very, very least, the lede should be simple and comprehensible to EVERYONE who might come across this article. Ford MF (talk) 18:20, 31 July 2008 (UTC)
 * We can talk about it, but please keep technical off article pages. I've moved it here to the talk page. --Trovatore (talk) 18:41, 31 July 2008 (UTC)
 * My bad. I didn't see that notation on the  page.  Perhaps the template itself should be amended to say that?  I'll know for next time.  Ford MF (talk) 21:31, 31 July 2008 (UTC)
 * Actually I just restored check talk to the template, so now it will warn you. There's been a bit of a back-and-forth about this; I'm not claiming that everyone agrees. It's my personal strong opinion that it should be kept off article pages. See the talk page of the template for a history of the arguments that have been made. --Trovatore (talk) 21:37, 31 July 2008 (UTC)
 * Aleph-0 is the smallest infinite number, aleph-1 the second-smallest, and so on. Aleph-0 is the same size as "1, 2, 3, ...".  Aleph-1 is bigger, but it's hard to say how much.  It's not bigger than the real numbers (0, pi, 7/3, the seventh root of 2, etc.), but they might be the same size.
 * If that helps you understand, or you think you can use this to improve the lede, great. CRGreathouse (t | c) 20:08, 31 July 2008 (UTC)
 * Well, I don't need a tutor. I just think the article is impenetrable, and therefore unhelpful, to a general audience.  I'm a little hesitant to try to improve the lede on my own, since I don't know the first thing about the subject.  I could certainly make it more understandable, but I'd have no way of knowing if my edits bore any relation to reality.  Ford MF (talk) 21:31, 31 July 2008 (UTC)
 * I tried to edit the lede myself, but couldn't manage to make it work so I brought it here (in different form). CRGreathouse (t | c) 21:44, 31 July 2008 (UTC)

Ford, can you make a more specific criticism? What's the first thing you find confusing, and why? --Trovatore (talk) 02:49, 1 August 2008 (UTC)

Story of Borges
El_Aleph, book of George Luis Borges, contains same-named story aleph, which is strongly inspired by concept described here. As there is title "Aleph in popular culture", maybe this story should be mentioned? Borges is well-known and very good writer. -- --90.191.147.186 (talk) 21:28, 3 February 2009 (UTC)

Choice Again
Okay, I could be very wrong, but do you need the axiom of choice to show that the union of a countable number of countable sets is countable? Given a bijection from the natural numbers to X1 and from the natural numbers to X2, one can define a define a unique bijection from the natural numbers to the union of the X1 and X2. Since the construction is unique, it must be constructive, and must not require the Axiom of Choice. Am I confused here? If not, then this article needs to be updated to remove that misconception. 24.136.26.148 (talk) 06:24, 9 May 2009 (UTC)
 * This argument gives you, correctly, that you can show without AC that the union of two countable sets (your X1 and X2) is countable. --Trovatore (talk) 07:01, 9 May 2009 (UTC)

See also the earlier discussion at Talk:Countable set/Archive 1. --Zundark (talk) 10:07, 9 May 2009 (UTC)

Choice a Third Time
"Using AC we can show one of the most useful properties of the set Ω: any countable subset of Ω has an upper bound in Ω. (This follows from the fact that a countable union of countable sets is countable, one of the most common applications of AC.)"

Why would anyone use the full AC when Countable Choice is sufficient? JumpDiscont (talk) 17:24, 30 September 2009 (UTC)
 * There are people who like to find the minimal fragment of axioms that suffice for various things (see reverse mathematics, for example) but for most mathematicians this is a secondary concern. AC is obviously true, so its consequences are also true.
 * That doesn't mean it's not useful to know that the result follows from countable choice. That is important; it shows for example that the result holds in L(R).  But it's not mandatory to emphasize this fact, in a context where people may not care. --Trovatore (talk) 17:33, 30 September 2009 (UTC)


 * When people say "Using AC" like that, they are usually referring to ZFC set theory, which does include the axiom of choice as an axiom but does not directly include countable choice as an axiom. &mdash; Carl (CBM · talk) 17:50, 30 September 2009 (UTC)


 * "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
 * — Jerry Bona
 * JumpDiscont (talk) 20:56, 6 October 2009 (UTC)