Relational databases and sql should be considered the pride and joy of computer science AND software engineering (with logic and math being the grand parents in this increasingly confused and mixed metaphor).
Politically, yes. SQL/relational wins the popularity contest. But so what?
Technically, no, it's not the center. It's merely the most popular, for the time being. This is a temporary state of affairs.
However, relational databases provide unprecedented ability to query information. Imagine one person puts data into a black box. 100 other people can ask it questions which were never intended by the original designer, and the system is able to answer those questions fairly efficiently.
The language used to query is orders of magnitude simpler than 'normal' programming languages.
If you query your database and the answer takes too long, you can add an index and the exact same query will start running more quickly!
Two different people can create two different tables. A third person can come in and 'join' these two tables.
I see nosql being a low level technology, higher than blocks/files/b-trees but lower than relational databases. Another way to say it is that relational algebra is the theoretical model while large parts of modern nosql tools are implementation details.
Not that most RDBMS conform to the relational theory 100% (or even 90%), but everything else is mere re-invention of the wheel badly.
As in: "Hey, let's trade ACID, security, uniform access to data by all apps" for cheap speed and ill-thought developer convenience.
No. RDF is formally proven via its Model Theory (as was KIF before it). That's arguably a stronger basis that relational "algebra"
Someone led you down the garden path.
There is absolutely no proof that RDBMS is objectively correct in any meaningful sense whatsoever. Did someone invent an arbitrary standard to measure it by, and then prove that it met that standard? Sure. But that's a far cry from a claim that RDBMS's are somehow "formally proven." That's just pure mathematical silliness and marketing propaganda.
Actually it's both technically AND scientifically.
Relational algebra is MATH describing the properties of relational data (that is, all kinds of data one would ever care to use), based on set theory.
All the other ways are just ad-hoc BS non-solutions, that happen to have this or that property because they forgot the tradeoffs.
It's irrelevant that you can concoct an arbitrary mathematical theory and then create a system to match. You must also prove that that theory is the most relevant and important way of looking at the data for the given problem at hand. RDBMS theory doesn't even bother trying to worry about the real world.
They only way to do this is if you have an ad-hoc faulty theory.