When (conventional) current flows through a diode from anode to cathode there is a voltage drop across the diode. This volt drop is fairly constant with only a slight increase when the current is increased.
What about reverse (zener) breakdown?
It is a bad idea to think of the normal operating behaviour of a diode as being short circuit or open circuit.
He's talking about an ideal diode, when it is fowrard biased the voltage across a diode (a real one) is +/-0,7V for a silicon diode, and it is fairly constant as JimB said, when it is reverse biased the current is so small (units of nA) that it is considered an open circuit, and you will measure any voltage (untill it reaches its breackdown voltage which is also the minimum voltage it needs to conduct in reverse ).
Zero to minus infinity simply means that the voltage *may* be anything between
zero and minus infinity. When the voltage across the diode is minus, the diode
does not conduct. A few examples:
v=-1, i=0
v=-2, i=0
v=-100000, i=0
So the voltage can be minus anything and the current will always be zero.
They seem to be saying that when the voltage across the diode is zero
it conducts (and the amount of current would depend on the external circuit).
They should have specified that 'v' is the voltage across the diode itself,
not the input voltage v to the circuit as it is drawn.