What is the proper case in "в век(е)"?

Question

As far as I know, these expressions are correct:

• В двадцатом веке
• В век второй мировой войны

Why does the first expression use the prepositional case, whereas the second expression uses the accusative case?

Answer 1

"В веке" is for literal period designation, like in "В 15 веке нарождающийся буржуазный класс всё более настойчиво заявлял о своих интересах".

"В век" is figurative, it's not that there's a century we are talking about, it's like "the age", "в эпоху", like in "мы живём в век инноваций" which can be translated as "we live in the age of innovations".

Now imagine yourself asking a question why age there's "in" in the phrase "in the age of innovations". Well, it's of because this is how in English specific relations between objects is designated.

And next thing crucial for understanding of what's going on is to realize that there's no such thing like one to one correspondences between languages. Because modern English has no cases some of constructions with same preposition can be translated in some other language (including, by the way, Old English) into two constructions with different cases, two constructions with the same case and different prepositions, two constructions with no prepositions at all etc.

It something that just should be memoized - when you talk figuratively, it's, "в век" - аnd that's the way it is.