By overpowered, I mean that a model that allows the idea of something having infinite precision leads to apparent paradoxes.  For instance you get constructs like Gabriel's Horn that have greater surface area than volume, or the case of choosing a random real number between 0 and 1:  the probability of picking any <i>particular</i> real number is (approaches?) zero, yet some number will be picked.<p>But in the case of the Cantor argument specifically he seems to be encoding an infinite amount of information in one number simply by extending the digits after the decimal point.  It's almost like - cheating, or more specifically an "exploit" allowed by what I call reals being overpowered.<p>(Here I have to make an aside:  the value Pi, which is actually a computable number, can also encode any amount of information because any number will appear at some point in the expansion.  But you'd have to indicate where in Pi your number starts, and your index values would (<i>on avergage</i>) be so large as to consume at least as many bits of information as it would to simply state the number you want.)<p>In the physical world, when you do something like measure the voltage on a capacitor you can only measure it to within a certain accuracy &#38; precision.  If you wanted to make an analog computer that uses different voltages to represent different values, you'd be limited by the accuracy and precision which you could measure it.<p>Note that, even if the voltage is infinitely variable, your ability to <i>differentiate</i> those values is still limited, so you still do not have infinite precision.  (In Turing's paper on computable numbers, he makes the case that even if a human brain can have infinite states, there would at some point be a threshold of similarity at which the brain couldn't distinguish two similar states from each other.)<p>So my claim is that real numbers imply the ability to measure something to infinite precision, and that that is contradictory to physical experience.