Sandra Miesel, Chronology of Technic Civilization IN Poul Anderson, The Van Rijn Method, New York, 2012, p. 611.
Only one story, "The Saturn Game," (pp. 1-73) is set in the twenty first century and it describes exploration of the outer Solar System by fleets of spaceships although it also reveals that there is an Apollo University in a city called Leyburg on Luna, the Moon, in 2057. (Willy Ley was a real world pioneer of space exploration.)
Mirkheim reveals further that, in the twenty first century:
space habitats called the O'Neill colonies were constructed in Lunar orbit sixty degrees ahead of or behind the Moon, at the Lagrangian points;
whole new industries flourished within the colonies;
such industries revived free enterprise;
this revival thoroughly transformed economics, thought and life;
meanwhile, diverse Terrestrial societies merged;
the economic, social revival and the cultural merging together generated a new, "Technic," civilization.
In the twenty second century, a further change occurred, faster than light interstellar travel.