Definition talk:Pair Coding

I was trying to keep the naming pattern consistent with Definition:Sequence Coding when naming this page, as well as Definition:Finite Set Coding. Maybe Definition:Ordered Pair Coding would be slightly better? --CircuitCraft (talk) 00:32, 8 July 2023 (UTC)

All of these names are broad and general, and are associated with concepts which are highly specific -- even within the field of mathematical logic and recursion theory. For example, the existing material in this area discusses another method of encoding a number, which uses progressive powers of primes. Can't immediately see exactly where you're going with this exposition, but it has to be emphasised that we have already gone a considerable distance down the general route that ends up with the Godel theorems.

As usual, it is of course important to report what your sources say, because it is probably the case that the author has done some reading around the subject, and has already established a system of naming which is consistent with a more general approach, unless it is a really basic recursion-for-beginners which doesn't really bother to give things well-defined names, or whatever.

Hence your sources are your first port of call. --prime mover (talk) 06:58, 8 July 2023 (UTC)

And yes, having read through what I wrote, and looked at the content of these pages, it looks like I myself was the perpetrator of this state of events. My bad. Let's leave the rename templates in, and then the task is to do a deep dive into e.g. Boolos and Jeffries and do the job properly. --prime mover (talk) 07:02, 8 July 2023 (UTC)

I pulled the set coding, in particular, from a very, very short mention in passing in the Encyclopedia of Mathematics article for Enumeration Operator. The Cantor pairing function is well-established in literature.

I actually added a mention of the prime-powers representation (indirectly) on Finite Set Coding. The issue is that we want each object to have a unique, canonical encoding, and prime powers force a set into a particular order. I suggested requiring the set to be naturally ordered before sequence-coding it, but as always, there are more options.

Fundamentally, in recursion theory, almost any reasonable representation will end up working, and which one to pick is a question of aesthetics and convenience (see, for example, Karp's "Reducibility among Combinatorial Problems," in which the problem of graph encoding is solved by mentioning a few options, stating that they can be inter-converted, and moving on). We're going for a little more rigor than that, so I plan to simply construct the ones that I need, and prove inter-convertability if necessary.

As for the namings, and this applies to both Coding and Enumeration Operator, I'm not aware of any other contexts in which they are used, despite the odd name in the latter case. It may be fine to leave them as-is (I'm thinking of Coding indicating a natural number encoding, and Encoding for strings over a finite alphabet). --CircuitCraft (talk) 12:57, 8 July 2023 (UTC)

Good call. I'll keep an eye open on this, but carry on what you're doing. --prime mover (talk) 20:05, 8 July 2023 (UTC)