Should the tags (peano-arithmetic) and (theories-of-arithmetic) be merged?

Question

TL;DR: Should the tags peano-arithmetic and theories-of-arithmetic 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 peano-arithmetic $\to$ theories-of-arithmetic 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 peano-arithmetic 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 peano-arithmetic using this SEDE query. (At the moment it returns 195 results.) And you can check that the tag peano-arithmetic is indeed shown on those questions. This means that the tags have not been merged. You can also see that the tag theories-of-arithmetic 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 peano-arithmetic. (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.

Answer 1

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$\Delta_0$.