Talk:Aleph number

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Too technical again[edit]

I think this article is still too technical. I think CRGreathouse's explanation above helps a great deal.

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.

I don't want to edit this article as others are far more knowledgable. I think adding examples and possibly reducing the technical jargon would help greatly. Would anybody like to work on this? Thanks very much. -      Hydroxonium (talk) 13:32, 7 September 2010 (UTC)

Aleph Travel Theory...[edit]

I've heard a theory that goes like this: At any speed less than infinity, it takes infinite time to reach point infinity. Travelling at infinite speed gets you to point infinity instantly, but travelling at that speed means it take infinite time to travel to point Aleph 1, travelling at Aleph 1 takes infinity to get to Aleph 2, etc. Can someone mention this in the article? Thanks. 69.27.22.70 (talk) 19:12, 19 November 2010 (UTC)

I am afraid that that is too vague to be appropriate for the article. JRSpriggs (talk) 20:08, 19 November 2010 (UTC)


Infinity is not a number. Saying that it takes you infinite time to travel to point \aleph_1 while traveling at infinite speed is like saying: I drove from New York City to purple at 52 gallons per pound. It's total nonsense. — Preceding unsigned comment added by Decoyd (talkcontribs) 09:18, 17 March 2011 (UTC)

Why are you commenting here? JRSpriggs said all that needed to be said. — Arthur Rubin (talk) 14:35, 17 March 2011 (UTC)

Obviously I don't think that saying what the original post was "vague" is the same as saying it's just flat-out wrong, and that the problem is that concepts of distance and time being unbounded are being conflated with the issue of cardinality, in the same way that the issue of the limit of a function being infinity is often confused with issues of cardinality, which is discussed in other posts. — Preceding unsigned comment added by Decoyd (talkcontribs) 22:20, 17 March 2011 (UTC)

Fixed point of aleph[edit]

In this section it is mentioned that the limit of the sequence \aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}}, \ldots is a fixed point of the "aleph function". But the rest of the article suggests that the subscript on aleph should be an ordinal number, i.e., that aleph is a function from the ordinals to the cardinals, and not from the cardinals to the cardinals. So this sequence... doesn't make any sense. (Would it be \aleph_0, \aleph_{\omega_0}, \aleph_{\omega_{\omega_0}}, \ldots instead?) bbi5291 (talk) 18:48, 23 February 2011 (UTC)

Cardinals (well, Alephs, anyway) are also ordinals. If one were to emphasize the fixed point, it would \omega, \omega_{\omega}, \omega_{\omega_{\omega}}, \ldots — Preceding unsigned comment added by Arthur Rubin (talkcontribs) 18:58, February 23, 2011

This page is filled with numerous errors.[edit]

I will go through the article and enumerate what I can see for now.

Intro: The introduction can be confusing to non-mathematicians. The intuitive concept of 'size' breaks down with infinite sets. Even though the cardinality page is linked, it would probably be best to talk about measuring size with the notion of a one-to-one correspondence. This breaks up the universe of sets into classes wherein any two sets in a class can be put in bijective correspondence. Then the aleph numbers can be thought of as a canonical choice of representative from each class. Statements like "The cardinality of the natural numbers is \aleph_0..." doesn't explain anything when you don't understand what is meant by 'cardinality.'

Aleph-naught: No working mathematician would call aleph_anything a 'transfinite cardinal' --use 'infinite cardinal.' The statement 'A set had cardinality \aleph_0 if and only if it is countably infinite' doesn't mean anything. The definition of countably infinite is that it has cardinality \aleph_0.

The comments throughout the article about cardinality of sets without the Axiom of Choice are all pretty much nonsense. Without AC, there is no real notion of cardinality. You can still split universe of sets into the classes mentioned in the above section, but the \aleph's won't represent every class, and there will be 'incomparable' classes. You don't need any form of choice to prove that \aleph_0 can be injectively mapped into any \aleph_\alpha (the identity function works every time).

Aleph-1: Like in the introduction, "\aleph_1 is the cardinality of the set of all countable ordinal numbers" doesn't help if you don't yet know what \aleph_1 is. \aleph_1 is the set of all countable ordinals. \aleph_0 is the set of all natural numbers.

'Therefore \aleph_1 is distinct from \aleph_0.' This is part of the standard definition of the class of cardinals: \aleph_1 is the least ordinal >\omega that cannot be put in bijection with \omega. 'The definition of \aleph_1 implies ...that no cardinal number is between \aleph_0 and \aleph_1' seems unnecessary, and the comment about ZF without choice is, again, does not mean anything. You don't need AC to prove that the \aleph's are well-ordered. You just don't get that the \aleph's 'represent all the sizes.' Don't use \Omega to denote \aleph_1. It's not used very often; use \omega_1 and \aleph_1. I am fairly certain that you don't need AC to show that any countable subset of \omega_1 is bounded in \omega_1. If C\subseteq\omega_1 is countable, then it is well-ordered, so you can inductively define an order-isomorphism from C onto some ordinal \alpha. Then \alpha is countable... etc. I don't think this uses AC.

The second paragraph of this article isn't meaningful. What does 'This is harder than most explicit descriptions of "generation" in algebra' mean?

The Continuum Hypothesis: 'It follows from ZFC that the celebrated continuum hypothesis is equivalent to the identity 2^{\aleph_0}=\aleph_1.' No. CH is the statement that 2^{\aleph_0}=\aleph_1. 'It is not clear where this number fits in the aleph number hierarchy.' I think this statement is misleading and vague. You are about to discuss the independence of CH, so just leave it out.

Aleph-omega: Say that \aleph_\omega is the supremum or least upper bound of {\aleph_n:n<\omega}. '...moreover, it is possible to assume 2^{\aleph_0} is as large as we like.' This statement is vague. It depends on what you mean by 'as large as we like.' For example, we can't assume that the continuum is greater than the least strong limit cardinal.

Aleph-alpha for general alpha: You don't need choice to show that there is a next bigger aleph. But without choice, you shouldn't really be talking about the concept of cardinality.

I'd like to briefly comment on AC. There is a lot of confusion as to the meaning of AC. You don't need AC to prove that \omega and the Cartisian product of \omega with itself are in bijective correspondence. You can explicitly give the function. Or if you read the article on AC, you might be inclined to believe that if you want to show that for some given countable collection of countable sets, their union is countable, you must always use some form of choice. Again, this is not true. If you know what's in the sets, you may not need choice. AC tends to be necessary when you are dealing with arbitrary sets that you don't know exactly what's in them.

Also, in general, you need to be more explicit about the definitions of what you're working with. This is especially true when you are talking about set theory without AC. And, you don't want readers having to hunt down definitions of things like 2^{\aleph_0}. For someone who isn't familiar with set theory, the pages can look like a bunch of crazy symbols, and even if they are staring right at the answer, they may be blissfully unaware.

Perhaps this page should be folded into the entry on cardinality. I think the other page is far more accurate and polished. — Preceding unsigned comment added by Decoyd (talkcontribs) 09:12, 17 March 2011 (UTC)

I won't comment on readability, as I am (or was — I can't say I'm active, now) an expert on this field. The concept of cardinality does exist without choice; there isn't always a canonical choice of representative of each cardinality, or even a well-defined name of the cardinality within the system of set theory, but the concept of cardinality still exists. I'll use von Neumann–Bernays–Gödel set theory for notation:
By a canonical choice of representative, I mean:
(\exists F)( (\forall x)(F(x) \approx x) \and (\forall x)(\forall y)(x \approx y \rightarrow F(x) = F(y))).
(Or, equivalently, the equivalence relation "≈", treated as a class, has a system of representatives.
By a well-defined name, I mean:
(\exists F) (\forall x)(\forall y) (x \approx y \leftrightarrow F(x) = F(y)).
This is true in ZF and vNBG, but not necessarily in ZFU and vNBGU.
But you call still talk about cardinality, in the absence of choice, without either of those.
I quite agree that one can prove \alpha < \beta \rightarrow \aleph_\alpha < \aleph_\beta, and that there is always a next larger Aleph, without choice.
The use of \Omega for \omega_1 is nonstandard, but not really unusual. If you look at our article on Omega, you can see some of the other uses, even for ordinals.
The use of "transfinite cardinal" is ambiguous in mathematics; some use it for infinite cardinals, some for cardinals >= \aleph_0, and some for cardinals greater than \aleph_0. It's probably best not used here.
"\aleph_\omega is the supremum or least upper bound of {\aleph_n:n<\omega}" also requires some form of choice. I've actually worked with some details of a model in which that fails.
It's correct to say that \aleph_1 = 2^{\aleph_0} is one form of CH, but not really that it is CH. That's the same sort of thing as saying that Zorn's Lemma is the Axiom of Choice.
Perhaps, I'll have more to say, later. — Arthur Rubin (talk) 15:15, 17 March 2011 (UTC)
I thought that \alpha < \beta \rightarrow \aleph_\alpha \le \aleph_\beta, if \alpha \beta are infinite? -- cheers, Michael C. Price talk 15:57, 17 March 2011 (UTC)
No. \aleph_\alpha is always strictly less than \aleph_{\alpha+1}. I suspect that what you're thinking of is that it's consistent with ZFC that 2^{\aleph_\alpha}=2^{\aleph_{\alpha+1}} (at least, when \aleph_\alpha is a regular cardinal — I don't think you actually need that assumption for this particular result but things get a little more complicated without it). --Trovatore (talk) 18:46, 17 March 2011 (UTC)
(And of course my statement doesn't literally make sense as stated, because it seems to say "for all α it is consistent with ZFC that some property P(α) holds", which taken literally is nonsense, because α is an ordinal rather than a term and therefore P(α) is not a formula. But if you substitute for α anything obvious, it'll work.) --Trovatore (talk) 18:54, 17 March 2011 (UTC)
Yes, I was getting mixed up, exactly as you thought. Thanks for the clarification.-- cheers, Michael C. Price talk 19:11, 17 March 2011 (UTC)
Certainly cardinality exists with or without choice, but I agree with Decoyd that the AC-less parts of this article could use work. CRGreathouse (t | c) 19:20, 17 March 2011 (UTC)
Mostly trimming, I think. The article is called aleph number, which by definition refers to wellordered cardinalities. So the non-AC stuff is mostly out of place. It's on-topic to mention that, if AC fails, then not every infinite cardinality is an aleph number, but other than that I think the material is better treated elsewhere. --Trovatore (talk) 19:37, 17 March 2011 (UTC)
Do you have a particular 'elsewhere' in mind, or do you just mean deleted? CRGreathouse (t | c) 21:52, 17 March 2011 (UTC)
Hadn't thought about it in detail. In principle this sort of topic might be treated at cardinality. Whether any of the actual existing text fits there, I'm not sure (I haven't read it recently). --Trovatore (talk) 22:04, 17 March 2011 (UTC)

All the discussion of the concept of 'cardinality' appears to be that cardinality is the mathematical analog of the intuitive notion of size. Without AC, the parallels break down. Now, I don't know enough about NGB to say anything, but I can speak to ZFC. For those who may not be experts, perhaps a little background:

The splitting of the sets is the following: Let 'x~y' be shorthand for the formula 'there exists a bijection from x onto y.' This gives a meta-theoretic equivalence relation, i.e. 'For each x [x~x]' (reflexivity) and 'For each x,y,z, if x~y and y~z then x~z' (transitivity) are theorems of (much less than) ZF.

Now let me rehash a bit of what Arthur said above, when we talk about a cardinal class function, we talking about a formula in the language of set theory phi(x,y) (intuitively think of this formula as saying that y is the cardinality of x) such that the following are theorems: (1) 'For each x there exists a unique y such that phi(x,y),' (2) 'For each x,y,z[(phi(x,z) and phi(y,z)) if and only if there exists a bijection from x onto y],' and (3) 'For each x,y, if phi(x,y) then phi(y,y).' (1) is saying that phi is indeed a (proper class) function, (2) is saying that phi 'respects' the equivalence relation ~, i.e. phi is a meta-theoretic congruence relationship with respect to ~, and (3) is saying that phi picks only one representative from each equivalence class induced by ~. Then you can have the formula: 'y is a cardinal' just be 'there exists x such that phi(x,y).' I would also assert that to have a true cardinal function (in the sense that cardinals reflect the intuitive notion of size), you would also want to have the class of cardinals is well-ordered (or at the very least linearly ordered). The notion of size is about comparing different objects, and if the class of cardinals isn't linearly ordered then you can't always compare sizes of sets.

Now, we have the following formula in the language of set theory: Card(x)='x is transitive, well-ordered by epsilon (the set membership relation), and for each y in x and f:y\to x, f is not surjective.' Let phi(x,y)='(There exists f:x->y such that f is a bijection) and Card(y).' Then (1),(2) and (3) are theorems of ZFC. Additionally, you obviously get that the cardinals are well-ordered, which makes me happy.

In ZF alone, the above choice of phi(x,y) do not give you (1),(2) and (3), i.e. the cardinals in the ZFC sense don't necessarily represent all the classes induced by ~. I don't think anyone is disputing that. Perhaps Arthur Rubin knows more about the following, but I don't see how you could in ZF alone find a phi that works (and certainly not something that tells you anything about what the set actually looks like). It seems that it must require some form of choice or hypothesis beyond ZF, but perhaps Arthur could provide some references to help clear things up. And I think that discussions of quasi-set theories, while interesting, are way out of place in an article about the infinite cardinals.

When I say that without choice there is no real notion of cardinality, what I mean is this: You can define any property you want and call it 'cardinality,' but without choice you don't have a theory which allows you to compare the cardinalities of sets. In ZF alone, are you still just calling all the aleph's together with the natural numbers the cardinals? Are you redefining the term cardinal? In the first case, you aren't really talking about the 'size' of all the sets because some sets won't have a 'size.' In the second case, a new cardinality won't give you a theory in which you can compare an intrinsic property of a set to that of any other set. Maybe with ZF plus some other axioms, but again, you need to say what axioms and what you mean by cardinal. Either way, you've changed what it is that cardinality supposed to be formalizing. Certainly the mathematical world looks quite a bit different. I also think that a lot of subtle issues shouldn't be presented without explanation as it can really give the wrong impression to non-experts. For example, in this discussion, when Arthur states that there are models in which \aleph_\omega is not the least upper bound of the \aleph_n's, without further explanation, even those understand what each of those sets are may walk away with an impression that is quite far from the truth. Try explaining how there can be non-well-founded models of ZFC when one of the axioms of ZFC is the axiom of foundation. I've seen numerous fairly smart people try to understand things like this but come away with some very distorted picture. When someone is reading this without someone there to clarify, statements like this don't serve to enlighten, they just make things worse. It's not that we should hide these things, but present them in an appropriate setting.

With respect to \Omega, I can only say that you are right, the term is nonstandard. So if you mention it, put it as an aside, because you really don't run across it much. Moreover, if you go up to a mathematician (certainly any set theorist or topologist) and write the symbol \aleph_1 or \omega_1 down in front of them, there's high probability that they'll know what it means. If you just write \Omega, without saying what it means, you are likely to get quite a few blank stares. It's mentioned in this article as if it's standard notation.

With respect to CH, of course Arthur is right that it depends on how you define CH. I assert that the standard presentation of the statement CH is that 2^{\aleph_0}=\aleph_1. While historically, CH came out of a statement about the real line, CH (and more generally, GCH) is a statement about cardinal arithmetic rather than something particular to the real line. And if people are worried about the historical concerns, rest assured that Cantor himself ditched the real line to work on with the 2^\omega anyway. Also, ditching the real line as a starting place has numerous benefits, primarily that the average reader is unlikely to know exactly what the real line is anyway. Integers... for sure... rationals... probably, but the real numbers is usually presented to people as something like "all the numbers." You can move from CH to GCH with ease when presenting it formulated as a statement about arithmetic. Also, to someone just getting into the subject understanding CH as a statement about cardinal arithmetic makes it easier to see to the heart of the matter when it comes to it's independence: The Power Set Axiom really doesn't give us a lot to go on. You get a more succinct presentation which is more in line with the standard development, which I think is something that the mathematics project is striving for.

I strongly disagree that the ZF without AC just needs pruning. I think it should be cut out, to mention that things get weird and point someone to a more thorough discussion of what it means to drop AC. There are a lot of statements that are, at best, very confusing. The page still says "The definition of \aleph_1 implies (in ZF without choice) that there is no cardinal number between aleph_0 and aleph_1." While true, it gives the obvious implication that this a statement that might be at issue and that AC might have been an issue. Presumably we're working with the definition that \aleph_1 is the ordinal greater than \aleph_0 which \aleph_0 does not map onto (or something similarly close).

Sorry I don't know how to properly make all the fancy symbols. I'm also not trying to 'troll' these discussions. Overall, I think that if you want the wikipedia math pages really serve as a resource, things need to be built up in a far more precise manner. Think about the fact that the \aleph_\alpha's aren't even defined until nearly the end of the article. It's definition requires an understanding of ordinals, which isn't really mentioned. This page also is in deperate need of citations to authoritative sources. Best of luck sorting this all out. — Preceding unsigned comment added by Decoyd (talkcontribs) 02:26, 18 March 2011 (UTC)

One last commment: I think that people should be careful to also be clear about the difference between ZF and ZF + not AC. Even I above, have not been clear about when I'm talking about one or the other. There's a huge difference. — Preceding unsigned comment added by Decoyd (talkcontribs) 02:36, 18 March 2011 (UTC)

As you probably know, vNBG is a conservative extension of ZF, if you define the sets of the model of vNBG to be the sets of the model of ZF, and the classes to be those "things" defined by formula; if φ is a formula, then \{x | \phi(x) \} is a class. Converting my class function "F" to a formula φ in ZF, that would mean that the properties of a cardinality function would be:
(1) (\forall x)(\exists ! y) \phi(x,y) (aka, φ defines a function)
(2) (\forall x)(\forall y)(\forall z)(\phi(x,z) \and \phi(y,z) \rightarrow x \approx y)
(3) (\forall x)(\forall y)(\forall z)(x \approx y \and \phi(x,z) \rightarrow \phi(y,z))
(2) and (3) can be replaced by the counter-intuitive
(\forall x)(\forall y)(\forall z)(\phi(x,z) \rightarrow (x \approx y \leftrightarrow \phi(y,z)))
The additional properties of a representative function is that (2) is replaced by
(2A) (\forall x)(\forall z)(\phi(x,z) \rightarrow x \approx z)
or adding
(4) (\forall x)(\forall y)(\phi(x,y)\rightarrow \phi(y,y))
In ZF, a function φ satisfying (1), (2), and (3) can be informally defined as follows, using the rank function ρ defined in von Neumann universe. Define:
ρ(x) to be the least α such that xVα
Then we can define the "cardinality of" x, cn(x) to be:
\operatorname{cn}(x)=\{ y \in V_{\rho(x)+1} | y \approx x \and (\forall z \in V_{\rho(y)})(z \not\approx x)\}
Where the bound: \in V_{\rho(x)+1} is required to show that the set is properly defined in ZF. Even less formally, cn(x) is the collection of all sets of minimal rank which are equipollent to x.
Much of this should be in the article cardinality or cardinal number rather than aleph number, though.
"\aleph_\omega is a minimal upper bound of {\aleph_n:n<\omega}" is true without choice; that it's the least upper bound requires some form of choice. Similarly \aleph_{\alpha+1} covers \aleph_\alpha as cardinals, but does not necessarily strongly cover.
(definitions)
x is covered by y if  x < y \and (\not\exists z) (x< z < y)
x is strongly covered by y if  x<y \and (\forall z)(x < z \rightarrow y \leq z)
For examples on how to write these fancy formulas, see Help:Formula, although I never use \implies for "implies" except in modal or philosophical logic contexts. — Arthur Rubin (talk) 08:43, 18 March 2011 (UTC)

Forgive me if I'm being dense here, or maybe this is the problem and we're just talking past each other, but doesn't your cn function fail to have your property (4), which is my (3)? i.e. That cn doesn't pick a representative of the of `proper equivalence class' from the class? For example, if x=\omega, then rho(x)=\omega so cn(x)=V_{\omega+1}\setminus V_\omega. It's a trivial matter that in ZFC, cn(cn(\omega))\neq cn(\omega) (I'd wager this is provable in ZF alone, but this is irrelevant) so when phi(x,y) is the formula 'y=cn(x)' then for all x,y, if phi(x,y) then phi(y,y) is not a theorem of ZF, if not refutable. Weren't we assuming that a cardinal (in the sense that it is in the range of your cardinality function) should represent its own size? But perhaps I'm confused.

With respect to your comment that much of the contents of this discussion should be on another page, I would only reiterate what I suggested earlier. This whole page should be folded into the page on cardinality. What are the alephs but infinite cardinals? And beyond notational issues of what it means to write the symbols \aleph_\alpha, everything else is really a discussion about cardinality. Thanks for the link to the formula page. I'll take a gander. Decoyd (talk) 00:35, 19 March 2011 (UTC)

My "cn" doesn't meet property (4), but it's not necessary for \left| \left|X \right| \right| = |X|, for cardinality to be a well-defined subject.
And the cardinals which are alephs can be discussed without dealing with all cardinals, I don't see any reason why this article shouldn't stay. For example, there should probably be a mention of Hartogs number here. — Arthur Rubin (talk) 01:47, 19 March 2011 (UTC)

First, the entire discussion has been about picking a representative of each class. I talked about the aleph's being a canonical representative of each class. You stated in the next post that the cardinal class function F had the property that F(x) was in bijection with x, i.e. that you are picking a representative from the class and so on. There is a huge difference between picking something from each class versus only being able to pick a set of representatives. This difference is exactly parallels the difference between ZF and ZFC when it comes to sets. By this I mean, in ZF if you have an infinite set of non-empty sets, so you know there's stuff in there, but you can't (in general) actually work with things from those sets simultaneously, whereas with choice you can actually select one from each set. That's a HUGE difference. In the first instance, for example, you can't form arbitrary products of sets. Goodbye topology. (Nobody likes you anyway. =P) That's silly, so we throw in AC. I think here, calling what you've got above cardinality is a stretch. Think about how cardinals are used in ZFC: You want to determine the cardinality of an infinite set, you start trying to pair it with the alephs until you get a match; you can check for membership in a class. With this, that's impossible, because while you know you have lots of equivalence classes but you can't actually put your hands on anything in all of them. In ZFC, we can prove: For each x, there is a cardinal y such that x is in bijective correspondence with y. What's the analogue here? For each x, there's cardinal y such that x is in bijective correspondence each element of y? That may sound satisfactory at first glance, in a world without choice, you can't necessarily touch those bijections. You know it's in A class but you can't tell which one. It that to many it may appear to be just as good because they think of how cardinality is used in a world with AC. Even if you've considered all this and decided that you are still comfortable with the weakened notion, I believe that very few other readers of the article will have even understood there could be an issue.

In the end, I don't think we're going to resolve our differences, so I'm happy to agree to disagree, of course I'd welcome more insight from you. Despite all my bravado, I know very little about set theory outside the standard ZFC. But I hope that anyone still reading this will, at the very least, see that things aren't so cut and dry when dropping choice. It can get quite complicated. Arthur is a superb mathematician who knows this subject very well and at the start of this discussion he asserted that there was this cardinal function F such that F(x) is in bijection with x. I know that just because this 'cn' doesn't have this property, doesn't mean there is not such choice that will work. Just looking at something, even when you know the subject well, and saying 'Yeah, that looks good' isn't enough. I say absolutely ridiculous things all the time about my field of expertise, even when I'm trying to be careful. It's quite easy to say something incorrect or use a poor choice of words when you're just speaking off the top of your head. I takes someone of Arthur's expertise to sit down and carefully write up correct statements, stating what is meant by cardinality in the land without choice. I may disagree with him that his choiceless version of cardinality is good enough but his statements are accurate and I'm sure he's got a lot more to add. People who don't know the field as well as he does should be very careful before putting things on this page, even things that seem "obviously" true. How many people, after reading this discussion have read something that they hadn't considered? Just because something looks okay, doesn't mean it's correct. There are still a number other things, from easy fact-checking, e.g. I think Godel's paper on L was in 1938 not 1940, to more technical items, e.g. I still assert that you don't need choice to prove a countable subset of \omega_1 is bounded.

Regardless of if I'm right or wrong, all statements of fact are non-trivial and need citation. The best way to resolve any disagreements may be to just make everything precise and cited. Don't use technical terms that aren't defined, either on the page or on another page where a precise definition is givne, in which case that page should be linked. Some concepts being used can be slipped in, e.g. The page says that under AC_\omega then aleph_0 is smaller than any other infinite cardinal. Not only should the concept of cardinal under that form of choice be made explicit, but also cardinal comparison under those hypotheses. Under weak forms of choice, there can be incomparable cardinals in the usual sense. Even things that nobody is disputing, e.g. the list of examples of countably infinite sets are theorems and should be cited. Things like that are going to be easy to find citations, but as much as I love planetmath, I think think that's a poor choice to cite. If this 'wikimatics' is going to become an authoritative source, the contents need to come from unimpeachable sources. When using nonstandard things, like \Omega, at the very least, state it non-standard. If this page isn't folded into the page on cardinality, then I think this page needs to figure out what its purpose is. I'd always considered the \aleph's as just a notational issue --a way to denote the infinite cardinals-- and so it would fit in nicely as a section in the page on cardinality. If it's more than this, what is the purpose of this page? What makes the aleph numbers related to, but separate from, cardinality? When I first learned set theory, I was told the \aleph_\alpha notation was used to connote cardinality, whereas the \omega_\alpha notation was used to connote order-type or when using it as an index set. Although, this doesn't seem to be the case in the literature. I had been told, when going through Munkres, \Omega was to distinguish \omega_1 with the order topology from \omega_1 with the discrete topology. This, too, doesn't seem to be the case in the literature. Is this addressed in any of the related articles? Should it be addressed? I know that, as a starting place, Enderton's book on set theory is a decent resource for the more elementary issues. Of course, Kunen's book is practically the bible of ZFC set theory. I have heard from a number of well-respected mathematicians that Jech's massive book has numerous errors when it comes to the more esoteric things, although I don't know what. I say this as a word of caution. A few of them were quite disparaging. Kanamori's book The Higher Infinite as a number of interesting historical notes that can provide nice color to an article. For example, I think that was the book that I read that Godel chose L to name the constructible heirarchy to stand for "the Law." While this wouldn't fit in here, I think that historical notes like that can really add something to an article. Decoyd (talk) 21:58, 19 March 2011 (UTC)

You seem to be assuming a concept of "cardinal number" different than ours or the standard mathematical one. The assertion that cn(cn(x)) = cn(x) (to make clear what the cardinal operator is) is not part of standard mathematics. And the assertion that any two cardinals are comparable is equivalent to the axiom of choice. If you couldn't define cardinals without the axiom of choice, that statement would be a nullity.
That being said, much of this article should be moved to cardinality or cardinal number, both of which implicitly assume the axiom of choice, but neither makes the assertion above. — Arthur Rubin (talk) 23:28, 19 March 2011 (UTC)


Decoyd, I think you're new around here, right? And I think you can be a valuable contributor. But you need to acquaint yourself with the concept of TL;DR. --Trovatore (talk) 01:07, 20 March 2011 (UTC)

I'm fully aware of 'tldnr.' I think that kind of attitude, while it makes for a snappy retort, is ignorant. There are some complicated issues that require more than a three or four sentences to explain. Certainly, my comments could be pruned back, but I was trying to be clear about issues that are at the limits (beyond the limits?) of my knowledge of my understanding. If you can't be bothered to read it, I couldn't care less. I don't think I'll be joining this community; I only posted here because I had some students come to me with a bizzare questions about how long it takes to get to aleph_1 traveling at speed aleph_0. They told me they had read it here. If anyone is interested in fleshing out some of these issues and is looking for resources, a quick literature search on cardinality in ZF provided two interesting publications: http://users.auth.gr/~tzouvara/Texfiles.htm/size.pdf (Published in Studia Logica Vol 80 no 1 2005) This article offers a definition of a generalized notion of cardinal which, if I'm reading this correctly, assigns an cardinal in the ZFC sense to each set and can hold onto some of the desirable characteristics of ZFC cardinality, but as a trade-off, had to let go of the existence of a bijection from a set to it's cardinality. Card(x)=sup{|y|:y\subset x is well-orderable} where |y| denotes the ZFC cardinality notion. The other article, 'Some Cardinals in the Absence of the Axiom of Choice' by Halbeisen and Shelah from The Bulletin of Symbolic Logic vol 7 no 2 2001 gives the definition of cardinality that Arthur Rubin gave and redefined what it means to be an aleph (an aleph is now a cardinal in the new sense that contains a well-ordered set). The paper can be found at http://user.math.uzh.ch/halbeisen/publications/publications.html Decoyd (talk) 02:57, 20 March 2011 (UTC)

What you're missing is that you don't have to explain everything. Most contributors to this page have a very decent understanding of the basics. You can summarize. --Trovatore (talk) 03:03, 20 March 2011 (UTC)
I see that the article cardinal assignment describes much of what I was saying about the cn function, adding the fact that it's not really necessary, as many of the properties of cardinal numbers can be described by ignoring the difference between a set and its cardinality. — Arthur Rubin (talk) 15:33, 21 March 2011 (UTC)

Pronunciation (reprise)[edit]

Note that several classical mathematicians, such as Brouwer[1] in 1913, specifically said this was to be read "aleph-null". Aleph null continues to be far more popular than "aleph naught" (according to Google) and authors such as Sanchez, Wolfram Mathworld and even Martin Gardner specifically say the pronunciation should be aleph-null. (Sanchez and Wolfram admit that some say "aleph zero", which is also the most popular pronunciation according to Google.) I think it is clear that "Aleph naught" probably occurs from confusion with expressions such as "x-nought" for x0. So why does this article still give preference to the non-preferred pronunciation? --seberle (talk) 18:09, 13 September 2011 (UTC)

I say "aleph-null" most often, and sometimes "aleph-zero", but not "aleph-naught". So I agree with you. JRSpriggs (talk) 20:00, 13 September 2011 (UTC)


Working set theorists mostly say aleph-naught, and I think that's what we should use. --Trovatore (talk) 04:53, 14 September 2011 (UTC)
(Just by the way, never ever ever use Mathworld as a guide to usage or nomenclature. Their coverage is tolerable from a mathematical point of view, but absolutely crappy from the standpoint of terminology.) --Trovatore (talk) 04:55, 14 September 2011 (UTC)
Mathworld is just one of many examples. (And yes there are errors, but in my own experience, I have found found far more errors in Wikipedia than Mathworld.) I simply haven't found any reputable or classical uses of the term "aleph-naught". Google confirms that "aleph-null" is by far the most popular pronunciation (or "aleph-zero"). Historic math papers and modern writers both say the standard pronunciation is "aleph-null". There are many references for "aleph-null". Do you have any references for preferring the non-standard "aleph-naught"? --seberle (talk) 16:03, 14 September 2011 (UTC)
It isn't non-standard. It is the standard pronunciation among working set theorists. The reason you get more hits for aleph-null is because of popularizers (and possibly some in other languages).
Look, Cantor of course called it aleph-null. Cantor was speaking German. Null is German for zero. When translating into English, the natural thing is to use the standard expression for subscript-zero, which is "naught". --Trovatore (talk) 18:17, 14 September 2011 (UTC)
That would be natural, but the translators translated it "zero" or "null". Help me out here. I'm trying to find papers where it was translated "naught" or a reputable modern writer that says the pronunciation has changed in modern times to "naught". I'm sure you say "naught" but other mathematicians are saying "null". Who is in the majority now? I don't know, though it seems clear the early writers all said "zero" or "null". We need some references so we're not just depending on our own experience. --seberle (talk) 11:52, 17 September 2011 (UTC)
Such references are not likely to be easy to find, because no one really bothers to spell the words out. In writing you use the symbols. The pronunciation is really used only in speech.
I really don't think that "other mathematicians are saying null" — at least, not set theorists, who are the ones most directly involved with the concept.
Mathworld and Martin Gardner really don't count, at least don't count much; Gardner is a popularizer and Mathworld is ... idiosyncratic. (See for example Wikipedia:Articles for deletion/Regular number.) Your Sanchez is someone I never heard of, but he appears to be (i) in a non-English-speaking country, (ii) not a mathematician (or at least not writing primarily about mathematics), and (iii) he calls it aleph-zero anyway. --Trovatore (talk) 19:54, 17 September 2011 (UTC)

Cardinality of infinite sets vs cardinality of well-ordered infinite sets[edit]

Seems like the article is not consistently clear on when the axiom of choice (AC) is needed or not in the statements it asserts.

  • E.g. the lead implicitly assumes AC since it says that aleph numbers represent the cardinality of infinite sets (all and not only the well-ordered ones).
  • On the other side, section Aleph-α for general α explicitly defines aleph numbers to be cardinals of initial ordinal.

I guess this reflects a difference of point of view between set theorists that are more inclined to implicitly assume AC and consider that all cardinals have an aleph and logicians that care about when AC is used or not, but, at the end, how to get out of this ambiguity?

It seems to me that the best approach is to be systematically explicit on when AC is used. First because it would no loose any information. Secondly because it seems to me to be the solution acceptable by the majority of people.

The remark also applies to related pages (Von Neumann cardinal assignment, limit cardinal, ...). Compare also to the position adopted in this Encyclopedia of Mathematics. Hugo Herbelin (talk) 15:45, 10 March 2012 (UTC)

Well ordered[edit]

I generally agree with Trovatore that the article should begin in the common case, which is ZFC, and only treat the less common case of ZF lower in the article. I started a section to do that. — Carl (CBM · talk) 13:17, 13 March 2012 (UTC)

Hi CBM (and Trovatore), if you want to go this way, then, I suggest you replace "In set theory" by "In set theory (ZFC)" in the first sentence of the article, so that things are clear for everyone. Then, there is some work to do also in the core of the article since every subsequent section of the article makes an explicit distinction about what happens in the presence of choice or not, creating an inconsistency with a lead silent on the AC issue.
As a side remark, I don't understand the purpose of the last paragraph of the lead (about ∞). Is it relevant to the article? Hugo Herbelin (talk) 09:14, 14 March 2012 (UTC)
No, I don't agree with calling out a specific axiomatization. In "set theory", by default, the axiom of choice is just plain true. We can go into detail later about what is provable in particular axiomatic systems, including ones that omit choice. --Trovatore (talk) 18:27, 14 March 2012 (UTC)
Will it be clear for every reader of this article that, by default, the axiom of choice is just assumed to be plain true in the lead, if it is not said so explicitly? For instance, the article set theory to which the lead links does not say that "set theory" necessarily means ZFC. And, again, the rest of the article does not assume AC to necessarily hold, so why should the lead follow a different approach? My proposition to add "(ZFC)" is very light. For a reader not at the level of understanding what ZFC means, it will just be 5 characters to skip at first reading, while for a reader who knows what ZFC is this will alert him that the claimed statement needs AC to be correct. (For instance, it is a pity that I had to rediscover by myself the missing assumptions of the lead when I read it, while if it had be correct at first, I could have relied on it without any further ceremony!). You, how would you ensure the correctness of the first sentence without calling out a specific axiomatization? Hugo Herbelin (talk) 22:50, 14 March 2012 (UTC)
Someone who is just learning about ZF for the first time would not know that the notion of infinity there is different from that which he learned when studying real numbers. So it is necessary to mention that infinity (∞) to avoid unnecessary confusion. JRSpriggs (talk) 09:53, 14 March 2012 (UTC)
I understand that there could be a risk of confusion for some readers, but it is strange to read this in the lead, because it seems to come unexpectedly. What about a section "What alephs are not"? Then, this section could for instance also explain into more details the difference with the ordinals, the beth numbers, ...
Incidentally, shouldn't the link on "infinity" be towards infinity rather than Limit_of_a_function#Limit_of_a_function_at_infinity? Hugo Herbelin (talk) 22:50, 14 March 2012 (UTC)

Low resolution images[edit]

FWIW I found the use of low resolution images quite painful to read on a high res display. I am sure that ways to render mathematical equations exist, and at the very least, their symbols are all part of unicode. — Preceding unsigned comment added by 220.233.85.116 (talk) 08:26, 13 June 2012 (UTC)

Interwikis[edit]

Here is the problem with the interwikis, which has caused some bot confusion. We have only one "real" article on aleph numbers, including aleph 0. A few projects, including cswiki, have separate articles on aleph 0 and aleph numbers. So there are really two groups of articles, each of which should be interwiki linked and which should not be interwiki linked to each other. One group is on aleph numbers in general, the second is on aleph 0 only. The names of all the affected articles can be seen at User talk:JAn Dudík.

Although the bot is editing the hewiki interwiki, the problem is not hewiki in particular, it's the projects that have two different articles, such as cswiki. The following interwiki path will always cause trouble because it starts and ends at different articles of the same project:

cs:Funkce alef -> en:Aleph number -> he:אלף 0 -> cs:Alef 0

One of those three interwiki arrows has to be eliminated. The first and the third seem obviously right (the titles match very closely), so the middle one will have to be left out. — Carl (CBM · talk) 19:21, 3 September 2012 (UTC)

I downloaded the usual intewiki bot code and ran it on this page, and after reading all the pages this page interwiki links to, it reported that it saw nothing to change. Eliminating one or two bots individually will not solve the problem because lots of people run interwiki bots; to reduce headaches we have to keep the page in a state that the bot will not bother. — Carl (CBM · talk) 19:32, 3 September 2012 (UTC)

There is another solution — the Hebrew wiki should rename its article to the equivalent of "Aleph number" (and add information about other aleph numbers, if possible). JRSpriggs (talk) 04:27, 4 September 2012 (UTC)
I see that the target of the Spanish inter-wiki link is es:Alef dos. They have no general article, but they also have es:Alef cero and es:Alef uno. So this may also be a problem. JRSpriggs (talk) 04:37, 4 September 2012 (UTC)

My experiences with the interwiki bots have been largely negative, and frankly wish they just didn't exist. I believe there is template code we can use to tell them to leave articles alone when their interventions are problematic. The bluntest tool is {{nobots}} but I think there are finer-grained solutions.

I don't really agree with Carl that the structure of iw links should cater to the algorithm used by the bots. Words can have cover overlapping but not identical semantic zones in different languages, and this should not be an obstacle to making the best link possible. --Trovatore (talk) 05:11, 4 September 2012 (UTC)

Your wish may be granted in a (hopefully not too distant) future.—Emil J. 11:34, 4 September 2012 (UTC)
I haven't really digested that page, but at a quick glance I don't think it really addresses my point, which is more that we should privilege human judgment over botlike consistency criteria when structuring interwiki links.
One clear test case is vitamin A versus retinol. At en.wiki we have two separate articles, and the idea is supposed to be (although it seems difficult to keep the articles in this state) that vitamin A is about all substances that have the same nutritional function (definitiely including other retinoids, with the status of carotenoids being a bit of a gray area; some sources call them pro-vitamin A rather than vitamin A proper). The retinol article, on the other hand, is supposed to be about that one particular molecule, and I think it should be primarily a chemistry article (this is the part that is difficult to enforce).
This choice, whether you agree with it or not, is surely at least a defensible option. Many other WPs, however, have only one article. Some of them are called (the cognate to) vitamin A, and some retinol.
Now, by Carl's criterion, the en.wiki articles would have no iw links except to Wikipedias that make the same division. I do not think that makes sense. In languages that have only one article, both vitamin A and retinol should be linked to that article. And we should be able to tell the bots, this is the way we like it, so leave it alone. --Trovatore (talk) 19:46, 4 September 2012 (UTC)
    • ^ Brouwer, L. E. J. (1913). "Intuitionism and Formalism". Bulletin of the American Mathematical Society 20: 81–96.