Why does 0 resistance mean 0 voltage? According to Ohm's law,$$
\left[\text{voltage}\right]~=~\left[\text{current}\right] \times \left[\text{resistance}\right]
.$$Does that mean if there is no resistance, then there can be no voltage? Why is that so? If you make a circuit with no load, and no resistors, why does the voltage have to be zero?
 A: Do not assume that a lack of resistors connecting two points implies that the resistance is zero between those points. In fact, it’s the reverse: The conductivity is zero, the resistivity is infinite, the current is zero, and therefore the voltage difference is undefined. That is, you can apply any voltage you wish between two unconnected points. 
In contrast, if you connect a perfect conductor between two points (e.g., an idealized superconductor or circuit wire), then the resistance is zero between the points, the conductivity is infinite, the voltage difference is zero, and the current is undefined. Here, the current is determined by the rest of the circuit. 
A: Voltage is not 0 since the potential difference is determined by the source. In the limit of 0 resistance, it will be like completely turning on a faucet as opposed to letting it trickle. You get an increase in current keeping the p.d. roughly constant (assuming the source has a sufficiently large capacity). The "0 resistance" is what is commonly referred to as a short circuit in everyday circuits. The high levels of current will be evident even from the thermodynamic properties of the wire since there will be significant heating.
A: It is important to realize that equations in physics have physical meaning behind them. So mathematically yes, $R=0$ means $V=0$, but what does this equation actually mean?
This equation is for when we apply a voltage to an ohmic material to determine what current will flow through it. Therefore, we are already assuming the object has resistance. If $R=0$ then we do not use this equation, since it no longer applies.
This is what I mean by saying our equations have meaning. The resistance does not determine potential difference. The resistance determines the resulting current due to a potential difference.
A: No. $V=IR$ is only valid for an ideal resistor. For instance, an ideal inductor has $R=0$, but the voltage across it is $V=L{dI\over dt}\ne0$.
An ideal circuit with no load and no resistors does not have $R=0$, it has $R=\infty$. $I=0$, so $V$ is given by the indeterminate form $0\times\infty$, which means any value is valid.
