Here is a related well-known idea: in nonstandard analysis, a subset B of the real numbers is infinite if and only if its extension B* contains nonstandard numbers.
It might be equivalent, actually. Think of infinite bit streams as hyperintegers (e.g. the infinite sequence of 1s is the hyperinteger (1, 11, 111, 1111...), in binary if you like). Let E be the set of encodings of all integers, and all their prefixes. E is infinite, so E* contains a nonstandard element. I believe (here is where my nonstandard analysis fails me) that such a nonstandard element gives us an infinite bit stream which can't be rejected by the decoder.
A less serious but also related idea is the Frivolous Theorem of Arithmetic, which states that almost all positive integers are really, really, really, really large.
Followup to: What's a "natural number"?
While thinking about how to make machines understand the concept of "integers", I accidentally derived a tiny little math result that I haven't seen before. Not sure if it'll be helpful to anyone, but here goes:
You're allowed to invent an arbitrary scheme for encoding integers as strings of bits. Whatever encoding you invent, I can give you an infinite input stream of bits that will make your decoder hang and never give a definite answer like "yes, this is an integer with such-and-such value" or "no, this isn't a valid encoding of any integer".
To clarify, let's work through an example. Consider an unary encoding: 0 is 0, 1 is 10, 2 is 110, 3 is 1110, etc. In this case, if we feed the decoder an infinite sequence of 1's, it will remain forever undecided as to the integer's value. The result says we can find such pathological inputs for any other encoding system, not just unary.
The proof is obvious. (If it isn't obvious to you, work it out!) But it seems to strike at the heart of the issue why we can't naively explain to computers what a "standard integer" is, what a "terminating computation" is, etc. Namely, if you try to define an integer as some observable interface (get first bit, get last bit, get CRC, etc.), then you inevitably invite some "nonstandard integers" into your system.
This idea must be already well-known and have some standard name, any pointers would be welcome!