Talk:Epsilon number

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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)

$\gamma$ or $\gamma -1$

"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)

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)

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 $\epsilon _{0}$. If $\epsilon _{0}=\omega ^{\omega ^{\omega ^{.^{.^{.}}}}}$, is there a similar way to define $\epsilon _{1}$ based on lower ordinals like $\epsilon _{0}$ or $\omega$?Eebster the Great (talk) 19:04, 20 April 2008 (UTC)

Just as $0,1,\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\ldots$ approaches $\epsilon _{0}\!$.

So also $\epsilon _{1}\!$ is the limit of $\epsilon _{0}+1,\omega ^{\epsilon _{0}+1},\omega ^{\omega ^{\epsilon _{0}+1}},\ldots$ and also the limit of $0,1,\epsilon _{0},\epsilon _{0}^{\epsilon _{0}},\epsilon _{0}^{\epsilon _{0}^{\epsilon _{0}}},\ldots$.

For any ordinal $\alpha \!$, $\epsilon _{\alpha +1}\!$ is the limit of $\epsilon _{\alpha }+1,\omega ^{\epsilon _{\alpha }+1},\omega ^{\omega ^{\epsilon _{\alpha }+1}},\ldots$ and also the limit of $0,1,\epsilon _{\alpha },\epsilon _{\alpha }^{\epsilon _{\alpha }},\epsilon _{\alpha }^{\epsilon _{\alpha }^{\epsilon _{\alpha }}},\ldots$.

If $\lambda \!$ is any limit ordinal, then $\epsilon _{\lambda }\!$ is the supremum of $\epsilon _{\beta }\!$ for $\beta <\lambda \!$. 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 $\operatorname {hyper4} (\epsilon _{\alpha },\omega )=\epsilon _{\alpha +1}$. 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 $\omega <$infinity, $lim$_x->inf ^x$\omega =$^$\omega$$\omega$. 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)

History?

Who first introduced ε₀? Cantor? What aboutε₁ ? linas (talk) 01:36, 5 June 2008 (UTC)

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)

countability

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)

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)

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)

Move to "epsilon numbers"?

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)

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)

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)

I don't really think this is a good idea. I think ε₀ has a modest intrinsic interest (first place that Cantor Normal Form runs into a circularity, first discovered proof-theoretic ordinal) that doesn't really extend to the epsilon numbers in general. A few words here or at Cantor normal form should suffice for the latter. --Trovatore (talk) 19:43, 12 September 2008 (UTC)

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)

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)

Definition as an equation?

The article says that epsilon numbers can be defined by the equation

$\varepsilon =\omega ^{\varepsilon }$

However, this doesn't make sense to me. By analogy, we should be able to define $\alpha =\omega +\omega +...+\omega +\omega$ as the smallest $\alpha$ that satisfies the equation $\alpha =\alpha +\omega$, which clearly isn't true, since $\omega +\omega +...+\omega +\omega =\omega *\omega =\omega ^{2}$, and $\omega ^{2}+\omega$ is a distinct ordinal from $\omega ^{2}$. Am I missing something? The definition in terms of tetration, $\epsilon _{0}=\omega \uparrow \uparrow \omega$, makes much more sense to me. —Deadcode (talk) 20:16, 22 April 2010 (UTC)

It's not analogous. There simply does not exist an ordinal α such that α=α+ω, but there does exist an ε such that ε=ω^ε.

On the other hand, there does exist α such that α=ω+α (the smallest such α is ω²). --Trovatore (talk) 23:22, 22 April 2010 (UTC)

To Deadcode: You seem to be assuming that $\omega =1+1+\ldots +1+1$. However, that is false, and indicates that you do not understand what an ordinal is. It would be closer to the truth to say that $\omega =1+1+\ldots$ . Notice that it has no last element. JRSpriggs (talk) 21:56, 23 April 2010 (UTC)

I realize that it has no last element. I only typed it out that way because I was thinking of the definition of Graham's number, which is finite, hence my mistake. Incidentally, the Ordinal Number page on WolframMathWorld makes the same error (for example it has $\omega +...+\omega$) so I'm in good company!

Please accept this rephrased analogy:

By analogy, would should be able to define $\alpha =\omega +\omega +\omega +...$ as the smallest $\alpha$ that satisfies the equation $\alpha =\alpha +\omega$, which clearly isn't true, since $\omega +\omega +\omega +...=\omega *\omega =\omega ^{2}$, and $\omega ^{2}+\omega$ is a distinct ordinal from $\omega ^{2}$.

I understand the difference between $\omega +1$ and $1+\omega$; the difference is that $1+\omega =\omega$. However, as I understand it, $\epsilon _{0}=.^{.^{.^{\omega ^{\omega ^{\omega ^{\omega }}}}}}$, not $({({({({\omega ^{\omega }})^{\omega })}^{\omega })}^{\omega })}^{\omega }...$, so ${\omega }^{\epsilon _{_{0}}}$ should be a distinct ordinal from $\epsilon _{0}$. The key is that exponentiation has right-to-left evaluation, and I believe this has been overlooked and has misled people into believing that $\epsilon =\omega ^{\epsilon }$. —Deadcode (talk) 07:24, 24 April 2010 (UTC)

To Trovatore: Could you please explain why that is true? I understand that $\omega +{\omega ^{2}}=\omega ^{2}$, because evaluation of addition is left-to-right, but that doesn't extend by analogy to ${\omega }^{\epsilon _{_{0}}}$ because exponentiation has right-to-left evaluation. —Deadcode (talk) 06:57, 24 April 2010 (UTC)

Right-to-left or left-to-right doesn't matter for addition — addition is associative. Somehow you've gotten things backwards here. The fact is that α^ω is always bigger than α, but ω^α is not. I could go into a formalistic explanation, I suppose, but I think it's beside the point; somehow you've just gotten something backwards in your head, and you just need to write it down carefully and examine it, and I think you'll see it.

(Just by the way, and not meaning any offense, being in company with MathWorld is in my experience not really something to brag about.) --Trovatore (talk) 08:41, 24 April 2010 (UTC)

Regarding MathWorld, I was joking, not bragging. But this is probably not the best space in which to continue this discussion, since it would probably become longer than anything else in this discussion page. —Deadcode (talk) 17:22, 24 April 2010 (UTC)

To Deadcode: ε₀ is neither of the two things you mentioned. It is $\epsilon _{0}=\omega ^{(\omega ^{(\cdots )})}\,.$ More precisely, it is the supremum specified in the lead of the article with the rightmost exponentiations executed first. JRSpriggs (talk) 22:43, 24 April 2010 (UTC)

Yes, thank you. I have learned how to properly think about $\epsilon _{0}$ after discussing it on a math forum — I no longer have any issue with its definition. BTW, I knew that it was evaluated right-to-left, and that is what caused the problem for me in the first place — I thought it broke the analogy of the ladder of operations. Now I see that all of the limit ordinal operations take place from right-to-left. For example, $\omega =1+(1+(1+(1+...)))$ and not $(((1+1)+1)+1)+...$ or $(((...+1)+1)+1)+1$, and that is why $1+\omega =\omega$ (and so on, up the ladder of operations). This is what was the key in helping me understand $\epsilon _{0}$. —Deadcode (talk) 02:18, 28 April 2010 (UTC)

I really can't figure out what you mean by this. Ordinal addition is associative; the order of evaluation is irrelevant. If you have a wellordered sequence of ordinals and you want to add them up, you can do it all in one fell swoop — you just concatenate them all, and take the order type. --Trovatore (talk) 03:08, 28 April 2010 (UTC)

To Trovatore: That is only true for a finite number of additions. When you add $1$ to $\omega$, or $\omega$ to $\omega ^{2}$, addition is no longer associative. Notice that if $\omega =1+(1+(1+(1+...)))$, then $1+\omega =1+(1+(1+(1+(1+...))))$; all you see is an extra level of nesting out of an infinite nesting, and it's obvious that $1+\omega =\omega$. However, if it were true that $\omega =(((1+1)+1)+1)+...$, then $1+\omega =1+((((1+1)+1)+1)+...)$, and it would no longer be obvious that $1+\omega =\omega$. Notice that I wanted a deeper justification for the definitions of ordinal arithmetic; when I had a problem with $\epsilon _{0}$, it was because I was thinking in terms of $\omega +1=1+1+1+1+...+1$, and $\omega ^{2}+\omega =\omega +\omega +\omega +\omega +...+\omega$, and $\omega ^{\omega }*\omega =\omega *\omega *\omega *\omega *...*\omega$, which is an analogy that breaks at exponentiation (when constructing $\epsilon _{0}$) — but now I realize that this is a nonsensical way of thinking about it; it's like saying that $1-0.99999...=0.00000...1$. —Deadcode (talk) 22:59, 29 April 2010 (UTC)

No, you're wrong. Ordinal addition is associative, period. The expression 1+ω is precisely the order type of one single point followed by ω single points. --Trovatore (talk) 23:37, 29 April 2010 (UTC)

To Deadcode: I think you are focusing too much on the operations and not enough on the ordering of the elements of the set. JRSpriggs (talk) 05:29, 28 April 2010 (UTC)

To JRSpriggs: On the contrary. It's just that I wanted to visualize arithmetic operations on ordinals in a way that is isomorphic to the corresponding operations on well-ordered sets of those order types. It is impossible to visualize a set with the order type of $\epsilon _{0}$ from the top down. That is why I wasn't able to justify $\epsilon _{0}=\omega ^{\epsilon _{_{0}}}$ in terms of thinking about nested sets of natural numbers. However, now I realize that it's easy to visualize $\epsilon _{0}$ in a different way. Like any other ordinal, it is the order type of the ordered set of ordinals smaller than itself; for $\epsilon _{0}$, this is the ordered set of ordinals in Cantor normal form. This lends itself to a bijection showing that $\omega ^{\epsilon _{_{0}}}=\epsilon _{0}$. —Deadcode (talk) 23:05, 29 April 2010 (UTC)

I want to demonstrate some more of what I've learned since starting this discussion section, by describing a bijection that shows that $\omega ^{\epsilon _{_{0}}}=\epsilon _{0}$. I'm not studying set theory formally, so my terminology is going to leave something to be desired, but here goes. The bijection is this: Take set $C_{0}$ to be the set of all Cantor normal form ordinals (all ordinals less than $\epsilon _{0}$) in ordinal order. Define a set schema $P(A)$ as this: for every element in $A$, create a natural number element in $P(A)$, preserving order. Every natural number value of an element of $P(A)$ corresponds to $\omega$ raised to the power of the corresponding ordinal in $A$, multiplied by the natural number value as a coefficient. Define the set $C_{1}$ to be the set of all sets following the schema $P(C_{0})$, in reverse lexicographical order, where each set can also be thought of as an ordinal in Cantor normal form. $C_{1}$ is still the set of all ordinals in Cantor normal form; this shows that the order type of $C_{1}$ is the same as the order type of $C_{0}$, which shows that $\omega ^{\epsilon _{_{0}}}=\epsilon _{0}$. —Deadcode (talk) 00:26, 30 April 2010 (UTC)