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TL;DR: Should the tags and be merged? Is some clean-up needed before doing that? Or are there some reasons not to merge these tags?


In October 2020, a tag synonym $\to$ was created as a result of this discussion: Creating theories-of-arithmetic logic tag However, the two tags have never been merged.1

On one hand, I understand that merging tags should be used carefully - since the process cannot be reversed. At the same time, leaving the two tags in this state means that some questions tagged will not be returned in various searches.2 The issues with searching were mentioned also in jeq's comment to the answer posted by the moderator who created the synonym.

Since these two tags were synonymized based on the consensus of the community (see the voting in the linked thread) - it does not seem very likely that the synonyms might be cancelled. (Of course, if there is a realistic possibility that the synonym might be cancelled at some point, it would be better to postpone merging. If the two tags are merged, reverting to the original state would mean a lot of manual retagging.) EDIT: Comments posted by Joel David Hamkins suggest that there is some opposition to the synonym between these two tags - and, consequently, if the synonym should be cancelled, merging does not make sense either.


1You can find the questions tagged using this SEDE query. (At the moment it returns 195 results.) And you can check that the tag is indeed shown on those questions. This means that the tags have not been merged. You can also see that the tag shows different number of question on the active tab and on the newest tab. (At the moment, 326 and 131 questions.) If you want to see other synonyms which have not been merged, you can try this SEDE query. But I should point out that this includes the synonyms for deprecated tags - where merging would cause more harm than good; considering that those tags have never been cleaned-up. See also: What to do with the synonyms for the deprecated tags?

2It is not clear to me whether this is considered a bug or whether it is intentional. Some related posts: How to find a question by its old tag when that tag is synonymized with another, Tag synonyms and searching and What are differences between synonyms with and without merging from the practical point of view?

3To illustrate this, we might use this question: Iterated Gentzen: or, a Sith objection to the proof of consistency of PA. This question is tagged . (In chat you can also see what were the tags at the time of posting this question.) It is among the search results if you search just using a combination of tags. And the search using keywords and tags works if you use some tag which is not a synonym like here. But if you use the tag peano-arithmetic or the tag theories-of-arithmetic, this question is no longer among the results.

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    $\begingroup$ Yes, they should, that was the intention behind it all along. There also has been a suggestion to disambiguate between "theories-of-arithmetic" and "models-of-arithmetic" or the like, but this can only be done manually on a case-by-case basis, and anyway, the latter tag was apparently never created. $\endgroup$ Commented Aug 3 at 7:27
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    $\begingroup$ @EmilJeřábek Concerning the tag for models, you probably mean the Joel David Hamkins' answer and the discussion in the comments there. It is certainly a related tag (if it gets created again) - but that is a bit separate issue from merging (peano-arithmetic) and (theories-of-arithmetic). $\endgroup$ Commented Aug 3 at 7:31
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    $\begingroup$ Yes, this is tangential to the merging. I just commented on it for completeness, as it was also discussed on the linked meta page. $\endgroup$ Commented Aug 3 at 7:34
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    $\begingroup$ @EmilJeřábek Actually, as you can see in chat, I reminded him of this issue. His response was that he is busy at the moment to make a post on meta about this. $\endgroup$ Commented Aug 3 at 7:38
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    $\begingroup$ I do not support the merger. In my view, there are many questions about PA and models of PA, and they are not accurately viewed as being about theories of arithmetic. Sure, all PA questions and models of PA questions involve theories of arithmetic, but they also involve mathematics. Shall we also merge these tags into a "mathematics" tag? No. Let's have a PA tag of some kind. I would support PA or models-of-PA as a tag, and that would be the main tag I would be using for questions in this area. I rarely consider any other theory of arithmetic than PA and its extensions. $\endgroup$ Commented Aug 7 at 21:51
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    $\begingroup$ I had created the models of PA tag long ago (and I evidently a few times more when it was deleted/merged in the past). For me, I want the tag: models of PA. This is the name of the research area, and is how many researchers in that area describe what they are doing. Perhaps this is a New York bias, with the MOPA seminar and Kossak, Gitman, also Schmerl, Kirby, etc., but it is still how I think of the subject matter. $\endgroup$ Commented Aug 7 at 21:54
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    $\begingroup$ I haven't checked, but I suspect that most uses of the tag peano-arithmetic are about the models rather than the theories, and so it would make sense to me simply to rename peano-arithmetic to "models of arithmetic" if that is the tag that people want to contrast with "theories of arithmetic". (But my preference is still for the tag "models of PA" since that is the name of the research area.) $\endgroup$ Commented Aug 7 at 22:08
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    $\begingroup$ @JoelDavidHamkins To me what you write seems to be more argument against the synonym - which exists since 2020. Naturally, if there is a consensus that the synonym should not exist, then merging is not an option, either. In any case, it would be easier to find out what the MO community thinks about this if you posted your suggestions as an answer. (In such case it would also be easier to ask for clarification what precisely you would consider as a reasonable situation around these tags.) $\endgroup$ Commented Aug 8 at 5:05
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    $\begingroup$ About the tag (models-of-pa) - it is a related tag, but to me this seems as a separate issue from synonymizing/merging (peano-arithmetic) and (theories-of-arithmetic). I will remind that you have mentioned that you plan post a separate question about such tag. (And in the comments to your answer, several users were supportive of a tag related to models.) $\endgroup$ Commented Aug 8 at 5:10
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    $\begingroup$ I will add that I am only aware of 9 questions which had the tag (models-of-pa). In those cases the tag was removed manually - not merged. Perhaps moderators can confirm whether such tag was merged at some point - but I am unaware of that. $\endgroup$ Commented Aug 8 at 5:59
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    $\begingroup$ I think I had said my piece at the other answer you linked above. I was against the synonym in the first place, and also against the merger for the same reasons. If we will have only "theories of arithmetic", I shall want to create "models of PA" again and place it on those 9 questions and all the others that are about models of PA. I don't understand why the models of PA tag was ever eliminated in the first place, since it is so clearly needed, especially if "peano arithmetic" will no longer be available. $\endgroup$ Commented Aug 8 at 12:29
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    $\begingroup$ But it isn't just 9 questions that the tag would be suitable for, if you look at the search stats I had made on my other answer, since there are hundreds of questions about models of PA, if you search on "model of PA" you get over 500, and " PA " gives 1400. $\endgroup$ Commented Aug 8 at 12:39
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    $\begingroup$ I've been involved in past changes about this issue, but at this point I prefer choices be made by people directly contributing in these topics. $\endgroup$
    – YCor
    Commented Aug 8 at 20:39
  • $\begingroup$ I have voted to close this question. $\endgroup$ Commented Aug 25 at 20:55

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I am not happy with the tag situation that we have created for these questions.

There is a very active area of research called Models of PA, short for Models of Peano arithmetic. This phrase appears in the name of the main research texts of this area (Models of Peano Arithmetic by Richard Kaye, The Structure of Models of Peano Arithmetic by Kossak and Schmerl), and it is also the name of prominent research seminars in this research area, notably the MOPA (=Models of PA) seminar in New York, which is one of the centers of work in this area. This is the phrase that researchers in this subject use to describe what area they work in.

There are over 500 posts on MathOverflow using the phrase "model of PA" and nearly 1400 posts using " PA ".

Many of these posts are about the models of PA, rather than about the various weak theories of arithmetic, although the latter topic is of course closely related. To my way of thinking, the tag "theories of arithmetic" is not a good fit for many of the questions.

To my way of thinking, the main distinction to make is between Models of PA and what is often called Weak Arithmetic, a subject focused on the theories of arithmetic. There is a conference series called Weak Arithmetic Days, which used to run between Paris and New York. I think the two tags we need are "models of PA" and "weak arithmetic". My sense is that most people working in weak arithmetic, however, would also be happy with "theories of arithmetic".

I was opposed to the synonymy of "peano arithmetic" with "theories of arithmetic" and I regret the loss of the old tag "models of PA". I am against the merger for the same reasons. I think I had created the "models of PA" tag many years ago, but noticed several years ago that it had somehow disappeared, and so I had created it again, but then it got deleted or changed to "peano arithmetic".

I think we should undo the synonymy and rename "peano arithmetic" to "models of PA" or to "models of Peano arithmetic" as a synonym of "models of PA".

If the merger does go through, however, then I suggest we immediately recreate the tag "models of PA" (with synonym "models of Peano arithmetic"), and I would be willing to over time to retag questions with that tag.

Alternatively, it would be an acceptable compromise for me if the merger took place, but the name of the tag was "peano arithmetic" and not "theories of arithmetic", since the issue for me is that the subject of models of PA is not usually conceived as centered on the theories, but on the models of PA. We could have tag descriptions to indicate that weaker theories are on topic for this tag.

It could be part of the tag description that the tag "models of PA" includes the model theory of weaker arithmetic theories, since that is indeed what the subject called Models of PA includes. A typical pattern in the area is that a theorem is proved first for PA, and then also often for countable models of ZFC or for ZFC+V=HOD, and then people work harder to get the corresponding result for weaker theories such as I$\Delta_0$.

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    $\begingroup$ Perhaps have tags for "models of arithmetic" and "theories of arithmetic"? $\endgroup$ Commented Aug 9 at 4:15
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    $\begingroup$ Thanks for posting an answer! Considering the recent comment by YCor, I suppose he would not delete the tag (models-of-pa) if somebody created it now. However, the other suggestion (cancelling the synonyms, renaming peano-arithmetic) can only be implemented by a moderator. $\endgroup$ Commented Aug 9 at 4:22
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    $\begingroup$ Looking at your previous answer I do not see any comments from the mods there - they might have missed it, after all, it was posted three years after the question. Let us hope that some moderator might notice the new thread and comment how the mods see the situation. $\endgroup$ Commented Aug 9 at 4:22
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    $\begingroup$ @ChristopherKing Yes, that was a compromise mentioned in the previous discussion. $\endgroup$ Commented Aug 9 at 20:37
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    $\begingroup$ It should be "models-of-peano-arithmetic" rather than "models-of-pa". For two reasons: one is that for people remote to the topic, "pa" does not necessarily ring a bell. Second, when writing a tag to a question, each word makes the system propose the tag where it's a subword. So "peano" would propose the unabbreviated one, not the abbreviated. In general, it's better to avoid abbreviations as much as possible in tags. $\endgroup$
    – YCor
    Commented Aug 11 at 7:24
  • $\begingroup$ Although this was mentioned in chat, I will briefly add a summary here. In this answer there are two suggestions: Cancelling the existing synonym. This can be done only by a moderator. Creating a new tag for models. The names models-of-pa, models-of-peano-arithmetic, models-of-arithmetic have been mentioned. This can be done by a regular user. (But still, if we want multiple names which are synonyms of one of them, help from a moderator might be useful.) $\endgroup$ Commented Aug 16 at 9:51
  • $\begingroup$ I will add that (as far as I know) the tag models-of-pa was created and removed once. The tag models-of-arithmetic was created and removed once. In both cases, the tag was removed by the same user - he recently said that he does not plan to be involved in the related tags. $\endgroup$ Commented Aug 16 at 9:51

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