# On the tags (transfinite-induction) and (transfinite-recursion)

At the moment we have two tags and . These topics are rather close.

I can imagine a reasonable question which could be tagged with the tags , , , . Having only five spots for tags, this might become a problem at some point (at least for some questions.)

Transfinite induction and transfinite recursion are not the same, but these two topics are definitely close to each other. Basically the difference is that in one case we are proving some result by induction on ordinal numbers. In the other case we are proving existence of some object using this technique. (I have decided to be cautious and avoided using work "constructing" or "defining" some object, but for informal description they might sound better and make the distinction clearer.)

• Would it be better to have these two topics in the same tag? If we want to create one tag for both topics, what would be the name of the new tag? Or should we simply create synonym between the two tags in one direction?
• Should we use simply to tag questions about transfinite induction/recursion? If yes, then should there be a synonym from these two tags to the master tag (ordinals), or is it sufficient to remove the tags and explicitly mention in the tag-info for ordinals that the tag is intended also for these question?
• Or is the situation satisfactory as it is currently and it is better to keep two separate tags to make the distinction?

The tag was created in August 2015 and it has 46 questions.

The tag is older, first occurrences can be found in June 2011. The tag contains 65 questions.

• I would probably consider a synonym (transfinite-recursion) $\to$ (transfinite-induction) as a reasonable way to go. I think the topics are close enough to each other, yet still important to warrant a separate tag. But I'll wait a bit to see what other people think. (And later I might possibly post an answer with this suggestion, unless somebody does that before me or some arguments in this discussion show me that this is not a good idea.) – Martin Sleziak Apr 15 '18 at 8:09
• I should probably also mention that I have pinged the creators of both tags to let them know about this discussion. – Martin Sleziak Apr 15 '18 at 8:10
• Induction and recursion are different things, even if there are some (formal) similarities. Mixing them up tagwise will only help confuse those that are not clear on the distiction. – Andrés E. Caicedo Apr 15 '18 at 12:41
• @AndrésE.Caicedo Why not expanding your comment a bit and posing it as an answer? The way I understand your comment, you're for preserving the current status, i.e., leaving the tags separate. – Martin Sleziak Apr 15 '18 at 14:21

For the record, it won't bother me regardless of what's done about this.

I don't like the idea of making one a synonym for the other. It bugs me when people talk about "inductive" definitions - people don't talk about "recursive" proofs, but if they did that would bug me too. Structuring our tags so as to blur the distinction seems to me like a bad thing.

If we have too many tags my vote would be to replace them both by (transfinite-induction-and-recursion).

I'd also say that I don't think it's optimal to regard them both as somehow covered under (ordinals). Of course the connection is clear if one knows a little set theory, but even if one did not know that small amount of set theory one could still use transfinite induction; it's just an accident that ordinals give canonical representatives of well-order types, not really essential to transfinite induction.

Consider for example that post where I evidently created that tag. It seems like a worthwhile argument, more natural/simple than what you see in, say, Folland. As written it's accessible to someone who's never heard of ordinals -- those intervals are well-ordered, so if there's a counterexample there's a smallest counterexample, qed. If I were, say, giving the proof in a real analysis class, formulating it explicitly in terms of ordinals would require a major digression.

• Personally, I like the suggestion to have (transfinite-induction-and-recursion) tag. By a lucky accident, this tag name would have exactly 35 characters, which is the maximal allowed length. – Martin Sleziak Apr 15 '18 at 14:25
• If having both tags is really an issue, this suggestion is a good compromise. – Andrés E. Caicedo Apr 15 '18 at 15:02