t=sqrt(1-v^2/c^2)
If you believe my interpretation of this equation, then both black holes and the speed of light are measurements of maximum occupation of spacetime. Neither is/can be infinite.
To recall the example of the square rug in a square room, once the size of the rug is equal to the size of the room, the values that are returned are the same as the values for a black hole or an object traveling the speed of light. We never say that the room is infinitely dense.
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