Therefore the rigidity of things around us cannot be caused by the Pauli exclusion principle.
Not true. The necessary condition for rigidity (disregarding amorphous solids) is crystalline microstructure. Subject to that condition, the Pauli exclusion force is responsible for a significant fraction of the rigidity of the material. In non-rigid materials, the atoms (or molecules) have more freedom to get out of each other's way under application of a force, all the while obeying the Pauli principle.
As for volume vs. hardness, for rigid materials, if you accept that the Pauli force is making it much more voluminous than it would otherwide be, I would say that the same force is also making it hard.
You say electromagnetic interactions dissapear at any significant distance, yet they are responsible for holding materials together?
There is essentially no direct attraction between distant atoms, but nothing stops you from having a line of atoms, each of which attracts the next.
Consider the case of a piece of glass. I am saying that electromagnetic forces hold the glass together. But those forces only exist between atoms that are in close proximity. However one atom holds on to the next holds on to the next through the whole glass, and it acts like a single rigid object.
However if you take that piece of glass and shatter it, you've separated the atoms along the cracks and they do not attract each other. In theory you should be able to put the pieces together and the crack will mend. In practice you simply can't put the pieces of glass closely enough together. (But The Feynman Lectures on Physics explains how sliding a piece of wet glass past a piece of wet glass does result in small portions mending then breaking, resulting in scratches on the glass. The water is necessary to lift surface impurities to allow pure connections to form.)
The necessary condition for rigidity (disregarding amorphous solids) is crystalline microstructure.
And what causes that crystalline microstructure other than the pattern of positive and negative charges on the atoms involved, resulting in adjacent atoms being attracted to each other?
Anyways as I've said, I've had enough of your drawing out rounds with asking questions that I am sure you know the answer to. Therefore if your next response isn't rather extraordinary, I'll be leaving this conversation.
One last comment then:
And what causes that crystalline microstructure other than the pattern of positive and negative charges on the atoms involved, resulting in adjacent atoms being attracted to each other?
You must be thinking of an ionic solid such as NaCl, but there are many solids (iron and silicon come to mind) in which the atoms have no net charge, yet the material is a crystalline solid. My point is that there is more at work here than just electromagnetic forces, namely quantum mechanics. Similarly, refering to our original discussion, there is more than just electrostatics that is responsible for the hardness of crystalline solids.
And no, I was not just thinking of ionic solids. For instance take ice. A molecule of H2O has no net charge, but locally there is a pattern of positive and negative charges on the surface which results in a predictable crystal structure. Most molecules aren't that strongly polarized, but no matter what there is always some pattern of charges, and atoms will tend to line up so that positive meets negative and they attract. They will do this even when passing by quickly, hence the Van der Waals forces.
Consider for instance Sodium. If there were no force other than Van der Waals holding the atoms together in the crystal, there is no explanation for the fact that Argon, a significantly heavier atom, has much lower melting and boiling points.
A typical atom has a net charge of 0 with minor local variations. Therefore its attractive force drops off much faster than 1/r^2. And so electromagnetic attraction between atoms effectively disappears at any significant distance.
If the structure of an atom is "obviously" affected by the exclusion principle, why would the structure of materials, made up of atoms, make any difference?
It is possible for both rigid and non-rigid materials to satisfy the Pauli exclusion principle. Therefore the rigidity of things around us cannot be caused by the Pauli exclusion principle.
In fact, both Coulombic repulsion and the exclusion principle are involved in the hardness of materials. I believe the Coulombic fraction is larger than 50%, but the other part is certainly non-negligible.
Are you talking about volume or hardness?
You are right that the exclusion principle results in volume. See http://en.wikipedia.org/wiki/Pauli_exclusion_principle#Stabi... for verification. (That effect is much bigger than I had realized.) However it does not necessarily result in rigid things that are hard.
Consider, for example, dropping a bucket of fresh water into salt water. Does the fresh water stop quickly upon impact? Obviously not. Now consider dropping a rock in the dirt. Does the rock stop quickly upon impact? Obviously it does.
What is different between the two scenarios? The answer is the structure of the materials, which are held together by electromagnetic forces. Therefore it really is electromagnetic forces that quickly stop falling solid objects that hit solid things.