Paris is its own département
Aren't French départements more the equivalent of American counties? With the French regions being more the equal of American states?
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If Earth's hundreds of nations did remain sovereign in the 22nd, 23rd and 24th centuries, what would United Earth's duties, responsibilities, and restrictions be?
The sovereign national governments of Earth would each send a representative to the United Earth body. The Prime Minister or President (or whatever) of a nation might themselves select a subordinate directly, or simply leave the selection to their Secretary of State, or legislature. The representative to the United Earth world government from the United States might be titled the "Undersecretary for United Earth Affairs," or something like that.
United Earth's duties as a non-ruling world government /international entity, subsequent to the formation of the Federation, would be to decide upon and sending a representative (or representatives plural) to the Federation Council.
United Earth would also send a ambassador to the Federation, apparently there is both a representative and a ambassador. The UE would also receive the Federation's ambassador to Earth.
Prior to the formation of the Federation, the United Earth governing body would facilitate activities between Earth governments and interstellar species. Not just the Vulcans, but others (like the Denobulan).
Subsequent to the Federation, the UE would receive and send ambassadors directly with other Federation Members (we've saw this in TAS). And direct diplomatic relationships with political entities outside the Federation (we saw this is DS9).
In addition to diplomatic relations, there would be trade and cultural exchanges. UE managing Earth's collective interstellar affairs
Domestically, United Earth, as a limited international governing body, could be of use in diplomatic matters between sovereign nations here, the (limited) world governance would orchestrate and facilitate trade matters, disaster relief, and other activities.
Not only trade activities on Earth but also with incoming and outgoing interplanetary and interstellar trade as well. The UE could provide co-ordination and customs services.
Manage Earth's other off world activities, things like colonization, and ongoing relationships with former colonies. Not just interstellar, but also colonies within our own solar system.
The UE could assume the activities of the World Bank.
The UE would provide a Interpol (International Criminal Police Organization) like organization on Earth and on the colonies..
Off-world there will be a need for something like the US Marshall's Service..
There will also be a need for something like the US Coast Guard in space. Inside the solar system, on pathways between Earth and trading partners, and between colony worlds. After the establishment of the Federation it's possible that Starfleet could do this, but it might make more sense for it to be locally managed.
Prior to the Federation, United Earth would oversee the early Starfleet, this would be the NX-01 and Archer, and the rest of the fleet. After the Federation is formed, If Earth maintains a "Home Fleet" defense force, it would be managed through the UE governing body.
After these
relatively few things, the United Earth governing body would have
no powers what-so-ever. It would not have the ability to "actually govern" the entire Earth, it would be a governing body with a limited mandate that prevent growth.
The ability of the Earth's hundreds of sovereign nations to control the purse strings of the UE governing body will be one of the ways to ensure that it remains within it's internationally agreed upon boundaries.
Individual nations would (of course) be able to conduct interstellar activities independently of the UE if they wished, trade and colonies, and diplomacy. But it would make economic sense to do so through the UE.
If the various home worlds of the Federation had as many separate nations as we current do today, a future sovereign nation on Earth might need as many as 30,000 embassies to cover them all. An expensive possibility, best covered collectively through the UE.
