Talk:Aleph number

incomplete

Need to define what the plus sign that looks like an exponent is in aleph sub alpha on the right of the equals sign in the section on aleph dash alpha for general alpha. 100.15.138.239 (talk) 23:27, 10 February 2017 (UTC)

It is defined in the previous paragraph. --Bill Cherowitzo (talk) 23:37, 10 February 2017 (UTC)

Implications of new research?

Just ran across this article this weekend: https://www.scientificamerican.com/article/mathematicians-measure-infinities-and-find-theyre-equal/ which seems to imply that all infinite sets have equal cardinality. It's been way to long since I studied any set theory for me to tie that into aleph numbers, but it's probably worth a mention. 38.109.137.2 (talk) 15:12, 18 September 2017 (UTC)

The title of that article is clumsily worded. No, it doesn't say at all that all infinite sets have the same cardinality. It says that two particular infinite cardinalities, ones important enough to have been given names (${\mathfrak {p}}$ and ${\mathfrak {t}}$), turn out to be provably equal, which was contrary to the previous expectation.

This isn't the right place to go into detail. I'd love to say more at the mathematics reference desk, if you'd care to ask a question there. --Trovatore (talk) 16:57, 18 September 2017 (UTC)

See Talk:Continuum hypothesis/Archive 1#Malliaris and Shelah for a related discussion. JRSpriggs (talk) 05:25, 19 September 2017 (UTC)

The continuum as aleph_n

Maybe I am missing something essential, but it seems to me that the statement 'for any positive integer n we can consistently assume that $2^{\aleph _{0}}=\aleph _{n}$' cannot possibly be correct, because it would imply that all $\aleph _{n}$ are equivalent. — Preceding unsigned comment added by FRZH (talk • contribs) 01:49, 3 February 2020 (UTC)

Notice that that is "for any [one] positive integer", not "for ALL positive integers simultaneously". So while it implies that $2^{\aleph _{0}}=\aleph _{3}$ could be true or $2^{\aleph _{0}}=\aleph _{25}$ could be true, it does not imply that $\aleph _{3}=\aleph _{25}$ is true. See Easton's theorem for more information. JRSpriggs (talk) 07:55, 3 February 2020 (UTC)

I think that I finally got your point, but still the formulation in the article is confusing, in particular the phrasing "we can consistently assume". If I understand correctly what you mean is that WITHIN ZF theory, it is not inconsistent to assume $2^{\aleph _{0}}=\aleph _{n}$, for any given n. — Preceding unsigned comment added by FRZH (talk • contribs) 22:48, 30 March 2020 (UTC)

I think you have a valid criticism. The precise formulation (of this limited part of the claim) is that, for any n, it is consistent with ZFC that $2^{\aleph _{0}}=\aleph _{n}$. Also the "we" language is disfavored for reasons of encyclopedic tone; the sentence could be rewritten to avoid this.

The more general claim is a little harder to explain. We'd like to say something like, "for any ordinal α, with blah blah blah exceptions, it is consistent with ZFC that $2^{\aleph _{0}}=\aleph _{\alpha }$".

But we can't really say that, because it doesn't actually mean anything, at least taking "consistent" to mean "you can't prove a contradiction". That's because we can't talk about an arbitrary ordinal in the same language we use to talk about whether things can be proved.

The statement can be made precise by using a model-theoretic rather than proof-theoretic notion of consistency, but this requires special care to make sure the concept you come up with still reflects what you want. It's tricky to formulate in an elementary article. --Trovatore (talk) 23:31, 30 March 2020 (UTC)

A union of countable sets is countable. Why do you need the axiom of choice to prove that?

I am writing about the following sentence in parentheses which appears in the main text of this article:

"(This follows from the fact that the union of a countable number of countable sets is itself countable, one of the most common applications of the axiom of choice.)"

Sorry, as a non-expert in these matters I cannot see why one needs the axiom of choice here.

For a non-expert like me it seems that the fact that

(*) "a union of countable number of DISJOINT countable sets is itself countable"

has a trivial proof, with no need for the axiom of choice. So, it seems to me that one way to deduce that

(**) "a union of countable number of (not necessarily disjoint) countable sets is itself countable"

would be to first show that if A and B are countable sets, then their difference A\B (the set of elements of A which is not in B) is also countable and then to apply (**).

Here again, I cannot see where the axiom of choice is needed. If it really is needed then it would be a good idea to explain why, for the benefit of non-experts like me. Thank you.

I suppose that my question is equivalent to the following question: Do you really need the axiom of choice to decide whether or not a given(an explicitly given?) element of A is also in B ? If so, why?

79.180.250.201 (talk) 15:51, 15 May 2020 (UTC)

No, your (*) is not provable without (a weak fragment of) the axiom of choice. You don't need much choice, but you need a little bit, to pick one enumeration for each of your pairwise disjoint countable sets.

If you want more details on that, please ask at WP:RD/Math, which is the right place to ask and which could use some interesting questions like that. As to whether we should include something about that in the article — not sure; don't have time to think about it right now, but maybe someone else should. --Trovatore (talk) 16:30, 15 May 2020 (UTC)

According to countable set, "a set S is countable if there exists an injective function f from S to the natural numbers". Suppose T is a countable set of disjoint countable sets. And suppose that $S=\bigcup T$. What is the function f which maps S injectively into ω? More specifically, if g is the function which maps T into ω, is 5 in the range of f and, if so, which element of S maps to 5? JRSpriggs (talk) 05:39, 16 May 2020 (UTC)

Gee, I dunno. Can't you just choose some member of S for each integer? 67.198.37.16 (talk) 03:16, 25 March 2024 (UTC)

Aleph-naught vs. aleph-nought

I think that "aleph-naught" is a misspelling of "aleph-nought". It is not just a matter of American vs. British spelling. Both "naught" and "nought" are words, but "naught" means "nothing", and only "nought" is used to signify the mathematical quantity 0, which is what is relevant here. If there is agreement on this, feel free to fix the article. Ebony Jackson (talk) 04:22, 12 November 2020 (UTC)

I've always understood subscript-zero to be pronounced as "naught" with an "a", not just for $\aleph _{0}$ but in general (say, for a variable called $x_{0}$). --Trovatore (talk) 04:27, 12 November 2020 (UTC)

I tried doing a MathSciNet search for published articles whose titles contain aleph-naught or aleph-nought. The search for aleph-naught turned up naught! But there were several with aleph-nought:

• Goodearl, K. R. Directly finite aleph-nought-continuous regular rings. Pacific J. Math. 100 (1982), no. 1, 105–122.
• Ara, Pere Aleph-nought-continuous regular rings. J. Algebra 109 (1987), no. 1, 115–126.
• Ara, Pere Stable range of aleph-nought-continuous regular rings. Ring theory (Granada, 1986), 1–7, Lecture Notes in Math., 1328, Springer, Berlin, 1988.
• Wu, Tongsuo Exchange rings with general aleph-nought-comparability. Comm. Algebra 29 (2001), no. 2, 815–828.
• Li, Qisheng; Tong, Wenting A remark on aleph-nought-comparability over exchange rings. Nanjing Daxue Xuebao Shuxue Bannian Kan 20 (2003), no. 1, 1–6.

Ebony Jackson (talk) 20:23, 13 November 2020 (UTC)

Personally, I pronounce "naught" and "nought" the same way, though I understand that this may differ from person to person. Anyway, my question is about the spelling, not the pronunciation. Ebony Jackson (talk) 20:32, 13 November 2020 (UTC)

Here are some further references suggesting that when a subscript 0 is spelled out, it is preferable to write "nought" instead of "naught" (and this is not only a matter of British English vs. American English):

• As a rule of thumb, "nought" (or "aught") is preferred when dealing with numbers, while "naught" is preferred outside of math. -- Prof. Merrill Perlman, Columbia Graduate School of Journalism
• "nought" is also synonym with "zero" in British English -- Marius Alza
• Nought is conventionally used in British English for the number zero...In both British English and American English, naught is used in nonmathematical contexts to mean nothing. -- grammarist.com

Ebony Jackson (talk) 07:06, 21 November 2020 (UTC)

It rubs me badly the wrong way for some reason, but I have to admit you have your cites in order. --Trovatore (talk) 00:05, 22 November 2020 (UTC)

Polish space cardinality

Prior to July 16th, this page stated that "Every uncountable coanalytic subset of a Polish space $\,X\,$ has cardinality $\,\aleph _{1}\,$ or $\,2^{\aleph _{0}}\,$". The natural reading of "or" in this context is as "that is to say", which assumes the continuum hypothesis. I noted this and deleted the sentence.

On July 17th, User:JRSpriggs, reverted my edit, seemingly taking "or" in its strict mathematical sense. That is to say "every such subset has cardinality $\,\aleph _{1}\,$ or it has cardinality $\,2^{\aleph _{0}}\,$". That is technically correct, but only because every such set has cardinality $\,2^{\aleph _{0}}\,$" (which is stated explicitly in the referenced article, which uses c for $\,2^{\aleph _{0}}\,$). As such (even rephrased to remove the English ambiguity) this statement is misleading and belongs in the article about the continuum, aka $\,\beth _{1}\,$. (And certainly not under $\,\aleph _{1}\,$).

On August 6th, an anonymous user changed the sentence to "Every uncountable coanalytic subset of a Polish space $\,X\,$ has cardinality $\,\aleph _{1}\,=\,2^{\aleph _{0}}$" presumably because they read it the same way I did. It is now explicitly assuming the continuum hypothesis.

I'm deleting the sentence. User:JRSpriggs is welcome to revise the sentence and move it to somewhere appropriate. Please do not revert. --Rxtreme (talk) 18:30, 7 August 2021 (UTC)

Good removal. Not clear why this point belongs in the article about aleph numbers. --Trovatore (talk) 19:15, 7 August 2021 (UTC)

It may be appropriate to remove it on the ground of irrelevance to the article, but I did not like removing a sentence backed by a citation on grounds of nothing more than a misreading of what it said.

For example, if the continuum is $\,\aleph _{3}\,$, then the cited article could mean that no subset of the continuum of cardinality $\,\aleph _{2}\,$ can be Polish. This would be a non-trival use of the property of being of cardinality $\,\aleph _{1}\,$ and mentioning it here (which is our only article on $\,\aleph _{1}\,$) would be significant. JRSpriggs (talk) 02:13, 9 August 2021 (UTC)

Well, if the continuum is $\aleph _{3}$, then no coanalytic subset of R can have cardinality $\aleph _{1}$, either. The theorem is that any coanalytic subset of a Polish space is either countable or has full cardinality (that is, $2^{\aleph _{0}}$, the same as the original space). (By the way, this goes way beyond coanalytic, except to prove it for more and more complicated pointclasses, you need more and more large cardinals.) --Trovatore (talk) 05:04, 9 August 2021 (UTC)

@JRSpriggs: Strictly speaking this is off-topic, but just in case you're interested, this fact goes through the fact that coanalytic sets have the perfect set property, for which I think Borel determinacy is enough. Search for "unfolded perfect set game". More large cardinals give you more determinacy, and therefore the perfect set property for more and more complicated pointclasses. --Trovatore (talk) 01:10, 11 August 2021 (UTC)

Hmm, actually now I'm not sure. It looks like Borel determinacy gets you the perfect set property for analytic, not for coanalytic. It could be JR's interpretation was correct after all. --Trovatore (talk) 01:16, 11 August 2021 (UTC)

Your interpretation of the sentence was contorted. I think you're the only person to read it that way (I didn't make the following edit, which was clearly for clarification) and it didn't reflect the actual source (so clearly the author of that sentence didn't intend your interpretation). (Reading an article that proves P and writing "P or Q or R" in the Wikipedia article is misleading and absurd.)

It's moot at this point, but I wish you considered which reading is more natural, and what the reference says, before reverting my edit.--Rxtreme (talk) 05:30, 10 August 2021 (UTC)

Actually JR might be right; this might be the most you can prove about $\Pi _{1}^{1}$ in ZFC alone. I wouldn't be shocked to find, say, that the reals of L were $\Pi _{1}^{1}$ in some forcing extension of L that falsifies CH but doesn't collapse $\aleph _{1}$. --Trovatore (talk) 01:53, 11 August 2021 (UTC)

Are you claiming that this is in the cited article? I don't see it.--Rxtreme (talk) 05:20, 11 August 2021 (UTC)

I'm not sure what you mean by "the cited article". The ref is to a book, one I don't have. --Trovatore (talk) 05:30, 11 August 2021 (UTC)

The preview available online does include, on page 20, the statement Proposition 1.4.17 can be generalized to show that each uncountable analytic subset of a Polish space has cardinality ${\mathfrak {c}}\ldots$. But note that it says analytic, not coanalytic. Because every coanalytic set is the union of $\aleph _{1}$ Borel sets, it would follow that the only possible cardinalities for a coanalytic set are countable, $\aleph _{1}$, or $2^{\aleph _{0}}$. I don't know whether that's all you can prove in ZFC or not. Just a little large-cardinal strength would eliminate the $\aleph _{1}$ possibility, unless of course CH holds. --Trovatore (talk) 05:51, 11 August 2021 (UTC)

The only mention of coanalytic set in the source is "In fact, Proposition 1.4.13 is a particular case of a deeper theorem in which the same conclusion holds under the more general assumption that the equivalence relation E is a Borel set (or even a coanalytic set) in X × X. A proof of this more general result is given in [145, Theorem 32.1]." There's nothing about aleph-null.--Rxtreme (talk) 21:37, 11 August 2021 (UTC)

The source is a book, so it's kind of hard to know what's not in it, but anyway I agree that the snippets we've discussed don't justify the claim. Even if it did, it wouldn't really be a fact about $\aleph _{1}$ as much as a fact about coanalytic sets. I don't support bringing the text back.

Still, I thought it was worth bending the talk-page rules to point out that some of the stuff said in this discussion wasn't really true, so that readers don't internalize misinformation, and to give JR credit where due. --Trovatore (talk) 21:50, 11 August 2021 (UTC)

Update: There's a relevant thread at WP:RD/Math#Coanalytic counterexample to the continuum hypothesis?, eventually to be archived here. I haven't checked all the details but, yes, it looks like all you can prove in ZFC alone for coanalytic is indeed "$\aleph _{1}$ or $2^{\aleph _{0}}$", and JRSpriggs seems to have read it correctly after all. I do not see any indication that this was justified by the claimed source (though again, I don't have the book; I was just going off the online preview), and even if it were, I don't think it would make sense in this article. I am almost tempted to go through the history and see when it was added and by whom; could have been someone who just plopped it in out of personal knowledge without updating the source. --Trovatore (talk) 16:31, 13 August 2021 (UTC)

definitions don't make sense without more context

If you didn't already know, how would you understand $\aleph _{\omega }=\sup\{\aleph _{n}|n\in \{0,1,2,...\}\}$? or $\aleph _{\lambda }=\bigcup _{\beta <\lambda }\aleph _{\beta }$? What is a supremum or union of cardinal numbers?

Yes, maybe you happen to know that we often identify a cardinal with its initial ordinal, and that we often identify an ordinal with the set of all smaller ordinals. But maybe it would be better to explain without this convention. And anyway, I don't see this convention explained at all in the article at the moment.

Yes, there is a link to least upper bound, but maybe it would be better to have at least a short explanation in place. And anyway, I don't see anything in the other article about cardinal suprema at all!

For that matter, what is $\aleph _{n}$ for finite n? Yes, $\aleph _{1}$ was explained as the cardinality of the set of all countable ordinals, so maybe a reader would guess what $\aleph _{2}$ means. But still, I think it wouldn't hurt to be clearer. --2607:FEA8:86DC:B0C0:B9B4:5D25:5007:EEE7 (talk) 23:31, 19 January 2022 (UTC)

Ah, for $\aleph _{2}$, I missed that there is also a link to successor cardinal. But again, maybe it would be better to have at least a short explanation in-place. And also, the link occurs in a later section of the article than the reference to $\aleph _{2}$. --2607:FEA8:86DC:B0C0:B9B4:5D25:5007:EEE7 (talk) 23:49, 19 January 2022 (UTC)

You may take this "The α-th infinite initial ordinal is written $\omega _{\alpha }$. Its cardinality is written $\,\aleph _{\alpha }~.$" as the definition, if that helps. JRSpriggs (talk) 20:02, 20 January 2022 (UTC)

It wouldn't hurt to articulate $\aleph _{2}$ as much as possible. This would give the reader a stronger sense of grounding. The current sequence of "0,1, trivial successor function that is obvious to all, omega" is not very intellectually satisfying. It feels like a cop-out or hand-wavey or I've-been-gypped-give-me-my-money-back feel to it. 67.198.37.16 (talk) 03:31, 25 March 2024 (UTC)

In popular culture

In Futurama s.2 ep.12 “Raging Bender” at 01:30 a movie theater is named Loew’s ℵ₀-plex — a pun on Multiplex (movie theater). Tuvalkin (talk) 08:13, 22 November 2022 (UTC)

A set is also a number?

"... the set of all countable ordinal numbers. This set is denoted by ω₁ [...] The set ω₁ is itself an ordinal number ..."

Is this language really correct? It sounds like it's saying the set is identical to a number, but how is that possible? 98.110.52.169 (talk) 08:21, 30 September 2024 (UTC)

This is a common convention in set theory. You identify an ordinal number with the set of all smaller ordinals (this gives you the so-called "von Neumann ordinals"), and you identify a cardinal number with the smallest ordinal having that cardinality (its "initial ordinal").

I actually agree that it's potentially confusing to dump this convention on readers without calling it out and explaining it. On the other hand, it may also be confusing to call it out and explain it. Some careful thought on this point is indicated. --Trovatore (talk) 18:27, 2 October 2024 (UTC)

It could be as simple as

The set ω₁, in the von Neumann definition of ordinals, is itself an ordinal number

though of course that's deferring the complexity to that other article. I agree with Trovatore that careful thought is worthwhile here. - CRGreathouse (t | c) 18:50, 2 October 2024 (UTC)