No one said the army of the CSA didn't exist, the question was, what is the best generic term for the army or soldiers, or even navy of the USA in this period, Union or Federal? I said clearly Union because it derives from the actual name of the government, Federal does not.
And Keith, clearly the end result of the War was a single unified government, called both before and after the conflict, the United States of America. The word Union describes the title of the country, the goal of the war, and the end result, a result that now has stood the test of time almost twice as long as the period between Washington and Lincoln.
As for the founding fathers' intent, that you mention elsewhere below, they were the most radical revolutionaries of their era. They were on the farthest left fringe of their political time. What made America "Exceptional" in Washington and Jefferson's era was their radical liberalism, a radical liberalism that made the conservatives of the colonial era, the Tories, flee to Canada to escape, when the Founding Fathers won their victory.
The tea partiers, the Sons of Liberty, whom our Founding Fathers cheered, and were card carrying members of, were a radical lefty Boston mob. The conservatives of their era, conservatives here in America stood against everything our Founding Fathers were for, and did everything in their power to prevent the dreams and goals of the Founders from becoming reality.
In Lincoln's time, the fervent hopes and prayers of the conservative crowned heads of Europe, the far right wing of their era, were for a Confederate victory, one that would signal the beginning of the end of the leftist liberal republic as a failed experiment, and forever weaken the growing power and influence, and more importantly the radical leftist inspiration of the United States. Ending the Union would work to the advantage of the aristocratic, grasping, globally ambitious monarchs of Europe.
The unchallenged strength of the Union after this War, destroyed forever the European hopes and made this Nation a few short decades later the incredible global superpower it is today. To say that no Union was achieved by the War I think is one of the most incredible declarations of non-fact that I have ever read.