Talk:Epsilon nought
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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)
[edit] γ or γ − 1
- "The γ-th ordinal α such that α = ωα is written
. These are called the epsilon numbers. The smallest of these numbers is ε0."
If the smallest is called ε0, isn't the γ-th called 
? Otherwise 
--SuneJ (talk) 07:34, 6 December 2007 (UTC)
- 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)
[edit] Definitions?
I'm just starting to look at transfinite numbers, and I'm wondering if there is a fully expandable definition of epsilon numbers other than ε0. If 
, is there a similar way to define ε1 based on lower ordinals like ε0 or ω?Eebster the Great (talk) 19:04, 20 April 2008 (UTC)
- Just as
approaches 
. - So also
is the limit of 
and also the limit of 
. - For any ordinal
, 
is the limit of 
and also the limit of 
. - If
is any limit ordinal, then 
is the supremum of 
for 
. JRSpriggs (talk) 02:11, 21 April 2008 (UTC)
Thanks, also, is this an actual mathematical limit we're talking about (a behavior) or is it actually a power tower omega (or some other transfinite ordinal) stories high? Another way of looking at it is, if the operation were defined, would this simply be omega tetrated to the omegath power? Eebster the Great (talk) 00:30, 10 May 2008 (UTC)
- Both. Using the order topology, this is the topological limit (is there any other kind?). If the hyper operator notation were extended to ordinals, we would get
. In other words, it is tetration as you surmised. JRSpriggs (talk) 21:14, 10 May 2008 (UTC)
-
- Thanks a lot. That's interesting, because although ω < infinity, limx->inf xω = ωω. And I was talking about a calculus limit, by the way, not really knowing enough about topology to discuss it one way or another. My understanding of topology amounts to the most cursory of all cursory explanations in the context of quantum physics. Eebster the Great (talk) 02:45, 15 May 2008 (UTC)
[edit] History?
Who first introduced ε0? Cantor? What aboutε1 ? linas (talk) 01:36, 5 June 2008 (UTC)
[edit] countability
The article states that ε0 is "countable", can this be made more explicit? I can see how ω2 is countable, but don't understand the leap to counting ωω, then extending this to ε0. linas (talk) 01:43, 5 June 2008 (UTC)
- The union of countably many countable sets is countable. --Trovatore (talk) 02:17, 5 June 2008 (UTC)
- 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)
