Talk:Ordinal number

Negative ordinals

A section quoting Russell in Principles of Mathematics was reverted as "surreal"! Now an illustration, as Venus has negative fourth apparent magnitude, is appended. Any comments? — Rgdboer (talk) 03:26, 27 November 2020 (UTC)

The "negative fourth magnitude" has no apparent connection to the topic of this article. It might have some connection with the ordinal numeral article.

(The "surreal" comment is probably a pun on the surreal numbers, which embed a copy of the ordinals, and which have negatives.) --Trovatore (talk) 04:01, 27 November 2020 (UTC)

At least the latter fact is worth mentioning in the article; and why not give Russell's reference (but not the full quotation) then, too? - Jochen Burghardt (talk) 10:00, 27 November 2020 (UTC)

Hmm, I suppose we could have a brief mention of the surreals (if we don't already; haven't checked) but the Russell thing is kind of anachronistic for that, maybe a precursor but not using terminology that has survived (come to think of it, you could say that about much of Russell's work in math logic). It's not really even clear from the excerpt given just what Russell was talking about -- does anyone have it handy to give more context? --Trovatore (talk) 19:09, 27 November 2020 (UTC)

Ok, I see. For now, JRSpriggs' recent edit (adding surreal numbers under "See also") is just fine. - Jochen Burghardt (talk) 19:26, 27 November 2020 (UTC)

By way of explanation: In scientific notation a x 10^N there is some discussion concerning the label for N beyond mere exponent. Some usage of order of magnitude has been noted but also that at the disambiguation page order (mathematics) nearly all the uses refer to a heirachial organization where N corresponds to an ordinal number such as at decade (log scale). A number of these orders of magnitude have been assembled in a WP:Category, and naturally with scientific notation there are negative Ns. Thus it seems necessary to acknowledge negative ordinals to properly document these order of magnitude articles in that category. — Rgdboer (talk) 01:35, 28 November 2020 (UTC)

@Rgdboer: are you sure you've understood what this article is actually about? I grant you that there's a different sense of the term ordinal number for which your comments might make a certain amount of sense; that sense is discussed at our article ordinal numeral, not here. That's fundamentally a linguistic rather than mathematical concept.

This article is about the ordinal numbers introduced by Georg Cantor. The interesting ones are infinite; finite ordinals are not in any essential way different from natural numbers. --Trovatore (talk) 04:33, 28 November 2020 (UTC)

Bertrand Russell wrote his book in response to continental developements in logic and set theory. How dare editors remove the on-topic WP:RS? Why are negative ordinals WP:Out of scope of this article? So what that the transfinite ordinals are the "interesting ones", the encyclopedia is to serve all readers, young and old. The article transfinite number is your focus, but the general reader needs to know that there are negative ordinals, especially for the reason given above. — Rgdboer (talk) 04:32, 29 November 2020 (UTC)

Russell's construction, in the generality of posets, would be in scope for articles on order theory, or (in that context) as a construction of the integers from the natural numbers, as ordered sets. It's out of scope for ordinals since the structures it produces are not regarded as ordinals (in mathematics, which is where he proposed it, right?) and nobody adopted Russell's idea of calling these things "negative ordinals". Russell misapplied or misunderstood the term "negative" here; the point in mathematics is to create a group from a semigroup, not a minus sign as a reverse direction. 73.89.25.252 (talk) 06:31, 29 November 2020 (UTC)

There are not, in fact, "negative ordinals", not in the sense intended in this article. This article is about a specific structure in set theory, which has no negative ordinals. Just because a reliable source uses the same word does not imply that it's talking about the same thing, or that it is on-topic for this article. --Trovatore (talk) 05:56, 29 November 2020 (UTC)

Should we disregard one old source by Russell or thousands of more recent articles which use "ordinal" in a sense which excludes negatives? JRSpriggs (talk) 06:04, 29 November 2020 (UTC)

In over a century, this definition of Russell's never caught on. You can "double" any linear (or partial) order that has a minimum element, putting in a reflection symmetry by fiat, but what does that accomplish? Apparently not much, since Russell's terminology did not gain acceptance. The gain in constructing integers from positive integers is create a homogeneous structure (transitive symmetry group), not an artificial left-right symmetry put in by hand by creating a duplicate mirrored copy. Russell's construction does not really sew the left and right sides together properly, in that whether you have unique successor or unique predecessor depends on which half you are in; it's not a single structure working differently than the sum of its parts. 73.89.25.252 (talk) 06:10, 29 November 2020 (UTC)

Faulty caption on pictorial?

There is an image with the caption "A graphical "matchstick" representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers."

It's probably my own confusion, but the caption makes little sense to me. Assuming a "matchstick" is a single vertical line, shouldn't the caption read "... ω^3 [not ^2]. Each stick corresponds to ... ω."  ? In any event, isn't the "+n" in the quoted caption useless? Jamesdowallen (talk) 00:23, 20 September 2021 (UTC)

I think, each stick denotes one number, so the sticks at the left end denote the numbers 0, 1, 2, 3, 4, ... . The leftmost "completely black triangle" (appearance to the human eye) then has ω at its tip. The next sticks right to it denote ω+1, ω+2, ω+3, ω+4, ... . The rightmost end of the picture then denotes ω^2. Maybe, the phrase "matchstick representation" should be explained in a footnote (along the above lines). - Jochen Burghardt (talk) 20:41, 21 September 2021 (UTC)

Get rid of pictures, or replace them

I am no expert, to be sure, on ordinals but can we please finally dump these attempts at graphical representations? the 'spiral' and 'matchstick' images -- they add nothing to the explanation and as far as I can see are not even discussed in the article text. 71.139.124.132 (talk) 13:20, 30 December 2022 (UTC)

The images are intended to illustrate the structure of (initial segments of) the ordinal numbers. The spiral form is just a hack to avoid an extreme aspect ratio. Do you have suggestion about alternatives? - Jochen Burghardt (talk) 19:07, 30 December 2022 (UTC)

The matchsticks are good. They clearly show how ω² can be embedded into the reals.

The spiral could be improved by defining clearly what normal function corresponds to one turn of the spiral. If I could, I would change it so that that function is α -> ω·(1 + α). However, I do not know how to edit the image. JRSpriggs (talk) 00:35, 31 December 2022 (UTC)

It seems that the Svg file can be edited with an ordinary text editor, so I can offer to change the labels according to your suggestions. However, I'm afraid I didn't understand them. It seems that after n turns, we are at ωⁿ; but I'm not sure whether your α corresponds to my n. - Jochen Burghardt (talk) 13:12, 31 December 2022 (UTC)

The outermost ring of the spiral should begin with 0 at the 12:00 high position and continue with the natural numbers coming closer together until the first turn is completed. At that point it should meet ω which should be directly under 0. The spoke beginning with 0 would continue with ω, ω², ω³, etc. until it reaches the center which would be ω^ω. The spoke beginning with 1 would continue with ω·2, ω²·2 ω³·2, etc.. Between these, new spokes should begin in the second turn, ω+1 would continue with ω²+ω. Clearly, most of these numbers would have to be replaced with just tick-marks and as you proceed around the spiral, and most of those would have to be omitted altogether. JRSpriggs (talk) 00:05, 1 January 2023 (UTC)

I started to redesign the picture according to your suggestion; it will take a few days to get the limit ordinals arranged. - Jochen Burghardt (talk) 17:19, 23 January 2023 (UTC)

It took me longer than expected, but now I came up with a first version that meets the above requirements (I hope), see File:Omega-exp-omega-normal svg.svg and File:Omega-exp-omega-normal.pdf. Comments are appreciated. - Jochen Burghardt (talk) 18:14, 8 February 2023 (UTC)

Thank you for your effort. Two problems:

1. Use ωn instead of nω. 2ω = ω < ω + ω = ω2 .

2. Please use darker colors. Yellow on white is virtually illegible (to me at least). JRSpriggs (talk) 01:00, 9 February 2023 (UTC)

I would use $\omega \cdot 2$ rather than $\omega 2$; the latter is unusual in my experience and the 2 looks like it could be a misplaced superscript or subscript. I agree about the yellow; that's pretty hard to see. --Trovatore (talk) 01:44, 9 February 2023 (UTC)

 Done. As for 1.: oops, sorry! As for 2.: I now darkened the colors from RGB (100%,0%,0%) for red (similar for green, blue) and (80%,80%,0%) for yellow (similar for cyan, magenta) to (80%,0%,0%) and (50%,50%,0%). I find the result still hard to read. On the other hand, I'd like to keep the "rainbow" effect which makes it easier to recognize which label belongs to which spiral turn. Moreover, inspired by the current image File:Omega-exp-omega-labeled.svg, I'd added some fade-out effect, e.g. at $\omega ,\omega +1,...,\omega +4$; should I turn that off? - Jochen Burghardt (talk) 12:08, 9 February 2023 (UTC)

Yes, I agree that File:Omega-exp-omega-normal.pdf should replace what is in the article. Thanks again. JRSpriggs (talk) 19:46, 9 February 2023 (UTC)

I think you put the wrong link. We don't want a PDF as an image (not sure that even works). Probably you meant File:Omega-exp-omega-labeled.svg. --Trovatore (talk) 01:52, 10 February 2023 (UTC)

No, Trovatore. That is not it. That is an old incorrect version. JRSpriggs (talk) 14:48, 10 February 2023 (UTC)