Talk:Epsilon number

message: Cantor's Transfinite numbers are also called aleph (aleph is the first letter of the hebrew alphabet, and slightly assembles a gothic N). Aleph zero is the first of these transfinite numbers, and the ordinal of the infinite countable sets, such as N (natural numbers), Z (whole numbers), and Q (rational numbers). Proved by Cantor with the diagonal proof, it's considered one of the most visual demonstrations..it is in fact included in what Paul Erdös called "The Book", a book in which were included the perfect proofs for Mathemathical theorems, alongside with it there is the subsequent Continuum Hypothesis, which is also the first of the 23 Hilbert Problems. This Continuum Hypothesis states that there is no set whose size is strictly between that of the integers and that of the real numbers, that is aleph zero and aleph one, respectively.
:What does this have to do with ε0? It is not an aleph; it's a countable ordinal. — Carl (CBM · talk) 00:30, 2 December 2007 (UTC)
thread_title:  comment

message: :"The \gamma-th ordinal \alpha such that \alpha=\omega^\alpha is written \varepsilon_\gamma. These are called the epsilon numbers. The smallest of these numbers is ε0."
If the smallest is called ε0, isn't the \gamma-th called \varepsilon_{\gamma-1}? Otherwise \varepsilon_0=\varepsilon_1  --SuneJ (talk) 07:34, 6 December 2007 (UTC)
replies: text: I added the clarification "counting from zero". Using an ordinal to specify the position of an element in a well-ordered set always counts from zero, but I agree that the formulation should not be ambiguous.--Patrick (talk) 09:32, 6 December 2007 (UTC)
thread_title:  <math>\gamma</math> or <math>\gamma-1</math> 

message: Who first introduced ε0? Cantor? What aboutε1 ? linas (talk) 01:36, 5 June 2008 (UTC)
replies: text: The ε notation has been around since at least 1908; Felix Hausdorff uses it in "Grundzüge einer Theorie der geordneten Mengen" with the modern meaning. I don't know if Cantor used the same notation but there is no doubt that he was aware of the concept. Mr Death (talk) 19:17, 12 September 2008 (UTC)
thread_title:  History? 

message: The article states that ε0 is "countable", can this be made more explicit?  I can see how \omega^2 is countable, but don't understand the leap to counting \omega^\omega, then extending this to ε0. linas (talk) 01:43, 5 June 2008 (UTC)
replies: text: The union of countably many countable sets is countable. --Trovatore (talk) 02:17, 5 June 2008 (UTC)
replies: text: See Ordinal notation#ξ-notation. Every ordinal less than ε0 can be described uniquely by a finite string consisting of just two symbols — "0" and "ξ". Clearly there are only countably many such strings. JRSpriggs (talk) 04:52, 5 June 2008 (UTC)
replies: text: Thanks, right. I was thinking.. well, I wasn't actually thinking about ordinals when I asked this question. linas (talk) 16:12, 5 June 2008 (UTC)
thread_title:  countability 

message: Given the recent change of focus, should the article be moved/renamed to epsilon numbers/epsilon number? After all, the article currently starts with the words “In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is…” — Tobias Bergemann (talk) 06:54, 12 September 2008 (UTC)
replies: text: I could be persuaded either way.  There's only so much to say about ε0 itself, and I think that what there is to say about it is better expressed in the context of "epsilon numbers" generally.  On the other hand, I suspect that there are more direct references to ε0 elsewhere in Wikipedia than to the general concept, and that the external world contains more references to ε0 without discussion of epsilon numbers generally than vice versa.  Michael K. Edwards (talk) 15:35, 12 September 2008 (UTC)

text: Either we should do this or redirect ordinal numbers to \omega_0.  (For the sarcasm-impaired, I am in favour.) Mr Death (talk) 19:27, 12 September 2008 (UTC)

text: Epsilon numbers after ε0 (i. e., the series ε1, ε2, ε3, ...) are not so interesting, but the idea of an "epsilon number" (a fixed point of an exponential map) is.  It seems to me that material that connects ε0 to the next "interesting" ordinal Γ0 belongs somewhere, and that no one who doesn't know the field is going to know to look for it under Veblen function.  And there's also the material about surreal epsilon numbers, which I think deserves a reasonably close linkage with the ordinal version given the common motivation (enumeration of fixed points of an exponential map) and the natural embedding of the ordinals in the surreals.  Michael K. Edwards (talk) 21:14, 12 September 2008 (UTC)
replies: text: I'm certainly not going to try to talk you out of being interested in what you're interested in. I submit that a lot of people who might be interested in reading about ε0 are not interested in epsilon numbers in general. --Trovatore (talk) 08:32, 13 September 2008 (UTC)
thread_title:  Move to "epsilon numbers"? 

message: I'm going to delete the section of "surreal epsilon numbers" dealing with the map x↦2x.  I deleted the same thing from the surreal numbers article.  Reason being, I can't find any reference for it.  There were no references listed (and there are none on this page either), and when I tried looking up the surreal exponential map myself, I found definitions that disagreed with the ones listed here.  I then asked about it on MathOverflow and nobody there had heard of this definition -- only the ones I had found in the books, which as I said earlier, is inequivalent.  I've tried to find User:Michael K. Edwards, who added it to both of them, but he appears to be long inactive and has left no way to contact him.  Thus, I deleted it from that page, and I'm going to delete it from this page now. Sniffnoy (talk) 04:37, 7 February 2013 (UTC)
thread_title:  Surreal epsilon numbers and the exponential map 

message: I'm a physics student, and upon typing epsilon nought, was redirected to this page, which was very annoying. Because the vacuum permittivity constant is probably in much higher demand than this pure mathematics, I have created a disambiguation page; I believe that ε₀ should also point to this disambiguation page. Ε₀ (disambiguation)
:Hmm, honestly, I can see your point.  But foo should never point to foo (disambiguation); that's just silly.  If we adopt your approach, then it's ε₀ itself that should be a disambiguation page.  This needs admin assistance — we would first move ε₀ to ε₀ (ordinal) (or whatever disambiguated title is chosen), delete the redirect left behind, move ε₀ (disambiguation) to ε₀, and then do some cleanup. --Trovatore (talk) 03:40, 18 November 2013 (UTC)
replies: text: How do we get an admin? Vacuum permeability works exactly like this; type in a name one thinks should direct them to vacuum permeability, and one gets vacuum permeability.  — Preceding unsigned comment added by Feynmaniac (talk • contribs) 05:49, 18 November 2013 (UTC)
replies: text: You could look at :Category:Wikipedia administrators and try to find an administrator with whom you are acquainted. Or just pick one at random. JRSpriggs (talk) 07:21, 18 November 2013 (UTC)

text: The way issues like this are usually solved is through WP:RM. Keφr 07:47, 18 November 2013 (UTC)
thread_title:  Redirect? 

message: There is some discussion at Wikipedia talk:WikiProject Mathematics#Epsilon zero. I started it as a discussion about the redirects Epsilon zero, Epsilon 0, Epsilon naught, Epsilon nought, ε₀, and ε0, but it later took a turn towards the name and scope of this article. — Tobias Bergemann (talk) 08:47, 8 January 2014 (UTC)
thread_title:  Discussion at Wikipedia talk:WikiProject Mathematics 

message: says stuff about gamma numbers in terms of ω^β and ω^ω^β, but what is β in this context? a successor ordinal? 24.137.229.197 (talk) 05:04, 19 March 2024 (UTC)
thread_title:  last part of intro section