Talk:Cantor's isomorphism theorem

Cantor's isomorphism theorem has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
Review: June 11, 2023. (Reviewed version).

GA Review

Passed. —Kusma (talk) 13:45, 11 June 2023 (UTC)[reply]

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

This review is transcluded from Talk:Cantor's isomorphism theorem/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: Kusma (talk · contribs) 13:52, 6 June 2023 (UTC)[reply]

Planning to review yet another of your maths GA noms, shouldn't take more than a few days. —Kusma (talk) 13:52, 6 June 2023 (UTC)[reply]

Section by section review

Lead: Is temporal logic more important than all the other related results?
- It's in the lead because it's an important application, not just a related result. The lead does discuss another application, Cantors' characterization of the order type of the reals. —David Eppstein (talk) 21:55, 10 June 2023 (UTC)[reply]
Shouldn't the date of the proof (and perhaps Hausdorff's name) also be in the lead?
- Ok, added. —David Eppstein (talk) 21:55, 10 June 2023 (UTC)[reply]
Statement and examples: The claim that the open interval (0,1) is unbounded is going to confuse readers who have seen "bounded" in the context of subsets of the real numbers. Some of the sources call the property "without endpoints", which might be clearer for the beginner. Could you explain (perhaps in a footnote) that you use the order theory definition of boundedness (which is not inherited by subsets), not the more familiar one from the real numbers?
- Copyedited to attempt to clarify this point, and explain the difference between the notion of boundedness here and the different notion in bounded set. —David Eppstein (talk) 22:03, 10 June 2023 (UTC)[reply]
Any example for the rationals and a subset of the reals should give a monotone increasing continuous function (by density), and any monotone increasing function on the reals will induce an order isomorphism of the rationals to their image. Are there any other interesting known such monotone functions other than Minkowski-?
- Density within itself is not the same as density within the reals. For instance $\mathbb {Q} \cap {\bigl (}(0,1)\cup (2,3){\bigr )}$ is an unbounded dense countable order but the identity function from it to itself does not extend uniquely to a real monotone increasing continuous function (there are too many ways to fill the gap), and any mapping to it from $\mathbb {Q}$ cannot be extended to a real continuous function at all (there will always be a jump discontinuity). As for more examples: even non-fractal ones can be interesting, for instance $x\mapsto x^{2}$ taking positive rationals to positive square rationals, but the question is whether any of these can be properly sourced. —David Eppstein (talk) 22:26, 10 June 2023 (UTC)[reply]
Thank you for pointing out the flaw in my intuition. —Kusma (talk) 13:40, 11 June 2023 (UTC)[reply]
The binary strings ending in 1 are a nice example (but of course they can be seen as just the dyadic rationals in (0,1) in disguise).
- Well, they're order-isomorphic to the dyadic rationals, in a very natural way, but that's not to say they're objects of the same type as each other. —David Eppstein (talk) 22:26, 10 June 2023 (UTC)[reply]
Proofs: "It alternates between the two orders for which one it searches for the first missing element" can you try to break this down a little more? As I understand it, we use the enumerations, take the first element that hasn't yet been used, find a partner for it, then do the same for the other order. "Earliest missing element" is a bit more temporal than "the one with the lowest index" or something.
- You seem to be asking about the immediately preceding sentence? That's the one that says how we find a missing element in one order, and match it to an element of the other order. The sentence you're quoting is merely intended to describe which order takes which role at each step of the process. Anyway, I rewrote this to clarify this point. —David Eppstein (talk) 22:36, 10 June 2023 (UTC)[reply]
textbook by Hausdorff: link and mention Grundzüge der Mengenlehre.
- Ok, added. —David Eppstein (talk) 22:43, 10 June 2023 (UTC)[reply]
"mechanized in Coq" what does that mean? Say what Coq is.
- Ok, expanded a little. —David Eppstein (talk) 22:49, 10 June 2023 (UTC)[reply]
Model theory: "The axioms can be formulated logically using either a strict comparison < or a non-strict comparison" that sounds more like a footnote, it is a bit distracting here.
- Moved into a footnote. —David Eppstein (talk) 23:08, 10 June 2023 (UTC)[reply]
The words "logical sentence" and "theorem" could be defined slightly more in their model theory meanings.
- Ok, added a gloss for sentence (at the start of the section) and theorem (towards the end). —David Eppstein (talk) 23:14, 10 June 2023 (UTC)[reply]
Related results: define "dense in each other"? (although there is of course only one thing it could mean).
- Rewrote to include an explanation, and also be a little closer to the source, which mostly describes it in terms of colorings instead of systems of disjoint orders. The source actually uses the property that each subset is dense in the whole set (not merely dense in each other) so I changed to match. —David Eppstein (talk) 00:13, 11 June 2023 (UTC)[reply]
"Suslin's problem... its truth" should be something like "The truth of this statement"
- Ok. —David Eppstein (talk) 00:14, 11 June 2023 (UTC)[reply]
Is there some context for Baumgartner's axiom?
- Ok. —David Eppstein (talk) 00:19, 11 June 2023 (UTC)[reply]

General comments and GA criteria

Took me a bit longer than planned due to RL and other wiki issues popping up (sorry). See above for prose comments, not really a lot to complain about other than the definitions being possibly confusing. Happy with scope and level of detail; source review to follow. —Kusma (talk) 17:03, 9 June 2023 (UTC)[reply]

Not thrilled by Chekmasov as a source (smart high school 10th grader). While Villani seems to read "Chalkdust", I couldn't see the connection to UCL. You could just use Bhattacharjee et al. for the definitions? (Some of them currently do not have citations).
- Bhattacharjee et al is good for the definitions. Chalkdust is helpful for sourcing some of the more obvious examples, the ones that a publication aimed at professional mathematicians rather than bright undergraduates might omit as too obvious. As for the connection to UCL, see the fine print at the bottom of https://chalkdustmagazine.com/ : "Chalkdust is published by Chalkdust Magazine, UCL, Gower Street, London WC1E 6BT, United Kingdom. ISSN 2059-3805 (Print). ISSN 2059-3813 (Online)." —David Eppstein (talk) 00:53, 11 June 2023 (UTC)[reply]
- That said, I found a better source (Dasgupta) for much of this definition/example material, and swapped out Chalkdust where I could. That currently leaves it sourcing only the statements that the integers are not dense, and that doubling is an order-isomorphism from integers to even integers. —David Eppstein (talk) 01:20, 11 June 2023 (UTC)[reply]
Use of Girgensohn checks out.
Marzion is not a reliable source. I don't think you need this "standard" either; what is in Silver should be enough to show that back-and-forth is common.
- Swapped out for an earlier and better source with the same quote; this also led me to a statement of elementary equivalence for arbitrary unbounded dense orders (with new source Langford). —David Eppstein (talk) 00:37, 11 June 2023 (UTC)[reply]
Lohrey-Matthissen checks out.
Worrell seems to be enough of a subject matter expert to use his lecture notes. For this (and other web citations) mentioning the publisher ("University of Oxford" in this case) would be nice but certainly optional for GA.
- Publisher added. —David Eppstein (talk) 00:24, 11 June 2023 (UTC)[reply]
Baumgartner checks out.

Other criteria:

No original research, no MoS issues I can see, no copyvio, two clearly free and relevant illustrative pictures.
Do you need the "Logic" navbox?
- Removed. —David Eppstein (talk) 00:24, 11 June 2023 (UTC)[reply]
A bit under-categorised perhaps; Category:Georg Cantor?
- Added, with Category:Theorems in the foundations of mathematics. Unfortunately Category:Isomorphism theorems seems to be about something else. —David Eppstein (talk) 00:24, 11 June 2023 (UTC)[reply]

First pass done, not many things again, just the two not so stellar sources. —Kusma (talk) 21:33, 9 June 2023 (UTC)[reply]

Happy now, nice article. Optional further improvements could include a better way to illustrate the proof, but I don't have a concrete suggestion. —Kusma (talk) 13:44, 11 June 2023 (UTC)[reply]

Good Article review progress box

Criteria: 1a. prose () 1b. MoS () 2a. ref layout () 2b. cites WP:RS () 2c. no WP:OR () 2d. no WP:CV ()

3a. broadness () 3b. focus () 4. neutral () 5. stable () 6a. free or tagged images () 6b. pics relevant ()

Note: this represents where the article stands relative to the Good Article criteria. Criteria marked are unassessed

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.