How would Earth's moment of inertia be affected if all people jumped simultaneously? We were arguing about this thought experiment with a friend. 
He was claiming that people continue to consist of earth's mass regardless of whether they touch the ground or not, whereas I claimed that when people lose touch of the ground, they momentarily cease to consist of the rigid body "earth". So, he concluded what when people jumped, earth's moment of inertia would increase, whereas I concluded that it would decrease.
Which is the right opinion?
 A: The answer is that you simply need to define what the Earth actually is.  If you define it one way, the moment of inertia will increase.  If you define it another way, it will decrease.
All the people in the world jumping is a pretty tiny activity.  The phrase "fleas on the back of an elephant" comes to mind.  It's easier to visualize this if we raise the stakes.  Instead of everyone jumping, lets have the entire mantle of the Earth get involved.  Really raise the stakes.  Have the entire mantle of the earth jump up a few feet, and we can worry about how to put things back together later.
If we define the Earth the way you did, the equivalent to this whole mantle jumping would be to say that the mantle ceases to be part of the Earth.  All that's left to be "Earth" is the core.  In such a situation, it's trivial to see that the moment of inertia must decrease.  The core has a far smaller moment than the entire mantle and core did.
However, if we define the Earth the way your friend did, then the mantle is still part of the Earth.  As such, there's more mass further from the center than there was before, so the moment has increased.
The paradox is not actually one of physics, but of linguistics.  The physics answer is that you are looking at the moment of inertia of two different bodies, so of course you will get different answers.  But linguistically, we like the idea that "the Earth" continues to be "the Earth," even after a radical change in topology.
Agree upon what "the Earth" means, and the answer becomes simple.
The answer probably depends on what you intend to do with the earth.  If you want to make it spin just a hair faster, then your definition is probably the more useful.  Atmospheric drag and the forces of people upon the dirt as they hit the ground will ensure the people continue to rotate at the same rate as the ground does.  Thus, it makes sense to keep them as part of the Earth, just as it made sense to keep the mantle as part of the Earth because it will soon settle and regain its position.  In such a situation, your friend is more right.
On the other hand, if you want everyone to jump and then have Superman spin the planet.  Spin it fast, fast, fast, faster until it would fling any human that touches it off into space (and thus also fast enough that the Earth will soon disintegrates under its own centrifugal pseduoforces), and I'm going to side with you rather than your friend.  In such a situation, the object that Superman must spin has a lower inertia because he doesn't have to spin up the people with it.  As such, you are more right.
It's all quite relative, for it's really all language.
