# Given a noise-free channel with limited bandwidth, why is the channel capacity infinite? [duplicate]

In the Shannon Hartley theorem, if N = 0 but B is limited, the theorem tells us that C is infinite. Why is this intuitively true?

If bandwidth is limited, I can only post so many pulses per second over the channel.

## marked as duplicate by hotpaw2, jojek♦Apr 5 '17 at 9:24

But if you insist: in a noise free channel, a single symbol could take one of $N$ values, with $N$ becoming arbitrarily large, because, due to the absence of noise, you could tell arbitrarily fine voltage differences apart. With that, a single (equally probable) symbol $x$ would have Shannon info $I = -\log_2 p_x = -\log_2 \frac1N = -(\log_2 1 - \log_2 N) = \log N\rightarrow \infty$ information.
But in terms of pure math, there is no ($\aleph_0$) finite limit on the amount of information that a single real number (one sample) can contain.