Well, you are quite right to call Steven in on his inference, but this does presuppose (since here we no doubt presuppose Church–Turing) that infinite streams are at least recursive, which is a nontrivial claim. Consider the incompatible claim that there are infinite streams in nature, but that these all are nondeterministic. It does seem to me to be defensible.

You forgot to mention that it is also a matter of faith whether or not there exists Infinite binary streams in our universe. I’m pretty sure there are none, and all binary streams are finite.

To give you an example (that is essentially the same as what I wrote above): suppose we are two banks that communicate credit card numbers to each other. We would like to agree on a protocol which allows us to “feed the input” and “read the output” but we impose no requirement on what we should be able to do with the credit card numbers. Then the following (bizarre) protocol is ok: whenever you want to communicate a credit-card number to me, just send me the string “hi”. You have fed the input and I have read the output.

As soon as we want to do something with the credit card numbers, this protocol will suck. And that is the whole point of my post: first figure out what it is that you want to do (such as “I want to be able to tell the digits of the credit-card number”), then you choose a representation.

You also ask about sufficient conditions that make a representation acceptable. This is a question that was studied by various people. For example, in Type Two Computation, “good representations” are characterized topologically and go under the name admissible. Another possibility is to express the abstract mathematical properties that characterize your data and then compute the representation from that using the realizability interpretation of logic and type theory. That is what RZ does.