You need to be very careful when you start throwing infinities around, especially in the real world/universe.

Density requires two factors; mass and volume, but the hypothetical singularities at the center of Black Holes have **no** volume, which isn't the same as infinitely small i.e. 0 ≠ 1/∞. Trying to apply the concept of density to the a singularity then, is not going to produce valid answers, and working on the basis that the density is infinite is really misleading yourself.

The problem stems in part from the mathematical slight of hand that allows to to work with zero as a valid number. For example, lets say I want to give you a gift of an apple, so I give you zero apples, tell you to take one for yourself, and then ask you to give the remainder back to me. Mathematically, this is fine, but in real life it's nonsensical.

With a BH, all we can say for sure is that space **behaves** as though there were a concentration of mass acting at that point, but not that there **is** a mass at that point. What we can almost certainly be sure of is that whatever is behaving as a mass at that point is not mass in any form that we know it and hence trying to apply density to it is equally invalid.