Season 3, not season 4. Season 2 had 2 parts, and people decided to call them two seasons even though it was a single seasonal run. I do not know why this is so hard for people and database sites to understand. They understood it with JoJo that season 1 was comprised of Part 1 and Part 2 in a single season. I once contacted TVDB about their numbering and they told me they only follow what official sources are doing. Crunchyroll has even cleared this up this season: https://cdn.discordapp.com/attachments/425634876641902603/1097354354622726224/image.png
Crunchyroll isn't exactly an official source despite being the official western license holder. As you see they label three different things as S2. Official Japanese sources outright do not have any listing that state what season is what. They simply use the subtext of the arc rather than seasonal naming.
As to whether it's Season 3 or Season 4... With Jojo it was literally announced that way, green-lit, and broadcast. With Demon Slayer however Mugen Train was green-lit separately... animated a year prior, then edited to tv format and formally aired prior to Entertainment District. At no point in time was it ever considered to be PART of the same season as Entertainment District.
In no scenario is labeling Mugen and Entertainment as a single combined season the correct path. So there really is only two legitimate options for it: don't call Mugen Train a season at all and instead list it as specials.... or call it Season 2 and move on. In which case the latter, which pretty much every database decided to go with for less confusion, would in fact make Swordsmith Village Season 4.
Hence aniDB and TheTVDB aren't likely to be convinced to change it and the only reason people in the anime scene bother to use SXXEXX naming is so that it agrees with and sorts based on these databases. Thus naming it S03EXX breaks the entire point.
I agree with you only on one point and that is another sensible way to do it would be to label Mugen Train as specials, which would still make Entertainment District season 2. However, it still makes less sense than considering both arcs as season 2 since it aired as a single cour during a single "season" (winter-spring-summer-fall segment).
The fact that Japan is not numbering it means that you cannot rely on the source to assign a numerical value, yet the west uses numbered seasons, so a company importing it has what I would call the most official say in what the numbering would be. I don't really like that appeal to authority all that much, though, and really fall back to the fact that it all aired together in a single seasonal run. Makes no sense to cut a single run up into 2 seasons.
To give more information about the ticket I filed with The TVDB in 2021 about this: their response indicated that they were going by the numbering used by the country of origin... which of course, we know Japan is not numbering them, so that response still makes no sense.
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