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There are quite a few questions on the main site related to (first-order) Peano Arithmetic and its fragments. Here are two examples: 1, 2 (but there are many others).

It would be nice, and feels appropriate, to have a tag devoted to them. Currently, they are tagged "logic", usually with an additional tag ("number theory", "arithmetic", "model theory", ...)

A dedicated tag would help locate these questions more easily, cross-reference them, etc. As a result, I believe it would certainly make them more useful in the long run.

I would appreciate suggestions for how to name this new tag (or reasons why such a tag is a bad idea or unnecessary). If we decide to go ahead with a new tag, there is then the question of whether some of the current tags should be removed from the questions, or if we simply add the new one to them.

If a consensus is reached, I can carry out the (re)tagging (just three or so questions a day for a few days, to avoid cluttering the front page).

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    $\begingroup$ We have both peano-axioms and peano-arithmetic, both of which ought to be merged into the new tag. $\endgroup$
    – François G. Dorais Mod
    Commented Sep 22, 2013 at 17:52
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    $\begingroup$ Also, the tag arithmetic ought to be deprecated since there are too many (wildly different) combos with arithmetic in them. $\endgroup$
    – François G. Dorais Mod
    Commented Sep 22, 2013 at 17:58
  • $\begingroup$ @FrançoisG.Dorais I agree. I was actually thinking of saying something about arithmetic in one of the questions we have on retagging, but it grew into its own thing. $\endgroup$
    – Andrés E. Caicedo
    Commented Sep 22, 2013 at 18:03
  • $\begingroup$ As a layman in these matters, isn't reverse mathematics a relevant tag? $\endgroup$
    – Felipe Voloch
    Commented Sep 22, 2013 at 19:08
  • $\begingroup$ @FelipeVoloch It depends on the question, it is certainly relevant in some cases, but I do not think it applies all the time. François or other experts may have a stronger opinion. $\endgroup$
    – Andrés E. Caicedo
    Commented Sep 22, 2013 at 19:11
  • $\begingroup$ @FelipeVoloch The traditional setting for reverse mathematics is second-order arithmetic whereas Peano arithmetic and fragments are first-order arithmetic theories. The two are, of course, strongly related but there are major differences. $\endgroup$
    – François G. Dorais Mod
    Commented Sep 22, 2013 at 19:45
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    $\begingroup$ It might be better to have two tags. "Bounded-Arithmetic" for very small theories which can be interpreted in Robinson's Q and one for stronger theories. $\endgroup$
    – Kaveh
    Commented Sep 23, 2013 at 5:09
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    $\begingroup$ One possibility for the second one is a tag containing "PA" but I think tagging questions which are not directly related to PA with such a tag might not be a good idea. Something like "first-order-arithmetic" looks more reasonable to me. On the other hand, we can use "first-order-arithmetic" to tag all of these questions. $\endgroup$
    – Kaveh
    Commented Sep 23, 2013 at 5:11
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    $\begingroup$ As one who does use these tags I agree with François G. Dorais: peano-axioms and peano-arithmetic ought to be merged into one tag. Actually I do not mind if that tag is the current Peano Arithmetic since I think that would naturally include fragments. The tag arithmetic ought to be deprecated. $\endgroup$
    – Colin McLarty
    Commented Sep 23, 2013 at 14:32
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    $\begingroup$ I merged peano-axioms into peano-arithmetic (until a better name is found) and I marked arithmetic as deprecated. $\endgroup$
    – François G. Dorais Mod
    Commented Sep 23, 2013 at 15:10
  • $\begingroup$ I think we should keep peano-arithmetic in any case, there are question which are about Peano arithmetic and it works well for them. For a more general tag I think first-order-arithmetic would be a good choice. $\endgroup$
    – Kaveh
    Commented Sep 24, 2013 at 23:09
  • $\begingroup$ Yet another tag name can be "subsystems-of-first-order-arithmetic" or ("subsystems-of-peano-arithmetic") but I think "first-order-arithmetic" would make then unnecessary. $\endgroup$
    – Kaveh
    Commented Sep 24, 2013 at 23:11
  • $\begingroup$ @Kaveh: Tags apparently have a maximum size of 25 characters: meta.stackexchange.com/a/13274/147362. So your latter two nice suggestions won't work. $\endgroup$
    – jeq
    Commented Sep 25, 2013 at 16:47
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    $\begingroup$ I am not sure to which extent this discussion is still relevant - but it might be worth mentioning that the limit for the length of tag names is now 35 characters. $\endgroup$
    – Martin Sleziak
    Commented Jan 31, 2019 at 16:24
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    $\begingroup$ Probably it is worth mentioning (at least in a comment) that the tag (peano-arithmetic) is now a synonym of (theories-of-arithmetic). The synonym was created in 2020. $\endgroup$
    – Martin Sleziak
    Commented May 1, 2023 at 14:03

1 Answer 1

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I suggest "weak-proof-systems". It is more broad than the situation of subtheories of a first order theory as mentioned by Andres Caicedo, but it conveys the notion of how much or how little a system can prove.

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    $\begingroup$ Would this be a tag someone with a question about a subsystem of Peano Arithmetic think of naturally? A tag they would consider? One they would search for? Otherwise there will be a lot of retagging and this will not be too useful. $\endgroup$
    – Andrés E. Caicedo
    Commented Sep 23, 2013 at 2:54
  • $\begingroup$ It is what I thought of, and is better than "weak" or "pa-fragment", in my opinion. However, there may be a more appropriate modifier than "weak". $\endgroup$
    – The Masked Avenger
    Commented Sep 23, 2013 at 3:07
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    $\begingroup$ "Proof system" suggests something different, in my opinion. $\endgroup$
    – Andrés E. Caicedo
    Commented Sep 23, 2013 at 3:09

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