[dr_dshiv]

Calculating Shannon entropy does rely on the symbolic alphabet and its probabilities of occurrance (see Wikipedia on Shannon entropy); however, I don't know how bandwidth is calculated. What makes you think it is as fundamental as temperature? One thing I know is that thermodynamic entropy has never been fully commensurated with information entropy -- unless someone has a ref otherwise.

Dialup internet did exceed 30kbps -- and that's because the bandwidth of the channel (the copper wire) was not the limiter. That's why DSL works and can in theory reach 1 gbps (https://en.m.wikipedia.org/wiki/Digital_subscriber_line)

I believe it is the channel plus the receiving mechanism plus the sending mechanism plus the encoding mechanism that determine the bandwidth. I am not asserting this, but that is my understanding.

[count]

Dial up internet used an analog transfer domain to encode the information. DSL does not, and is an entirely different technology (and is not 'dial-up').

56kbps modems and the like relied on digital telephone lines. I'm not sure where the 30kbps number comes from though - earlier pre-digital modems did go faster than that. Although you could argue those were 'pre-internet' as well...

[dr_dshiv]

Wait, but that's my whole point -- it's the same channel, but higher bandwidth.

[count]

If you're defining the 'channel' as 'local copper pair', then, sure?

But the other end of the local copper pair switched and became digital, which changes the channel in my eyes at least. There was then a series of bandwidth increases in the digital realm, giving significantly more efficient and effective use of the channel.

[gugagore]

It's not the same channel, because DSL requires equipment fairly close to the subscriber. You can't transmit DSL on telephone utility poles for miles, like you can dialup.