Btukfyl

One of the big issues I have run into while building the story I'm currently working on, is the fact that their are multiple nations that exist in an industrial state, while others are far past this point. Some countries are far ahead; with powerful medicine, efficient and sustainable energy production, and are living peaceably, balanced with nature, as space travel develops. Others remain at current levels of technology; viciously fighting for control, using cheap and inefficient energy sources, and over consuming their resources in pursuit of their own profit.

Is it possible for both of these societies to exist at the same time? How would this come about? I'm not entirely sure about what kind of history there would be between nations like that, but I'm pretty sure its not positive. Would that affect anything?

That's exactly our world.

On the same planet we have people using smartphones to order home delivered food from an etnic restaurant preparing food from a country located on the other side of the world, while other group of people still use wooden bows and stone arrows to hunt their dinner.

Each nation progress is a train moving at different pace, in some cases not moving at all.

Give a look at the relationships between Western countries and African sheperd nations or the Sentineleses to get a feeling of what this implies.

L. Dutch mentioned different levels of development in the real world. Can that pattern be translated to post-industrial vs. merely-industrial nations?

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  Environmental Balance:
  
  The industrialized nations have the means to really hurt the planet. By burning fossil fuel, releasing GM organisms, or starting a nuclear war, to name three possibilities. Will the post-industrial nations build carbon sinks, biowar defenses, and ABM to keep that in check, or will it be MAD, or what?


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  Global Travel and Commerce:
  A 19th century Australian aborigine or Chinese farmer had greatly diminished options compared to an European citizen. Even so, Chinese workers came into more industrialized countries. Unless there are attempts at embargo, won't post-industrial gadgets filter into the industrial nations, at least for the wealthy and middling classes?

There is a tabletop roleplaying game called Transhuman Space. In addition to space, various supplements detail the adventuring possibilities of an Earth with wildly different levels of development. (Disclaimer, I've known the writer for quite some time, but I wouldn't benefit personally in any way if you go and buy it.)

Absolutely. Technology progresses at different paces in different locations. This is due to a variety of factors, from environment, to population, to historical influences. There are isolated tribes scattered around the world today that are still living with Stone Age technology, and people still use flintlock muskets as their main hunting tool in parts of Africa. There is no reason why such a situation might take place in the future as well.

The post-industrial society will distort and deform the development of the industrial societies. A good example is how, in the real world, the industrial societies of the West deformed the feudal islamic societies like Arabia and the chinese imperial system.

For example, Saudi Arabia. The saudis are a feudal society but they have access to the products of advanced industrial societies because they have something that the industrials are willing to pay good money and, due to balance of power, can't simply invade and steal. So we have feudal institutions with advanced technology, the overlords controlling the faith of the people on social networks.

The saudis can't build the advanced technology they are using, they have to buy it from the industrial societies. They can't become industrial themselves because their institutions forbid that, they lack the social technology and aren't willing to pay the price of learning. At the same time, westernized saudis import western ideas to Arabia, ideas that are alien to that feudal society. Eventually some strange synthesis will occur, like the Islamic State: the ancient doctrines of the Ghazi mixed with SUVs and internet to spread the jihad.

Back to your scenario - The industrial societies will be deformed by the post industrial ones, depending on the needs and wants of the post industrials. The ideologies and philosophies of the post industrial will mix with the backward ideas of the industrial and strange and unpredictable ways.