Vs K unf vasvavgryl znal inyvq fhssvkrf, ng yrnfg bar bs K0 be K1 zhfg unir vasvavgryl znal inyvq fhssvkrf. Fb yrg k_{x+1} = 0 vs k_1 ... k_x 0 unf vasvavgryl znal inyvq fhssvkrf, naq 1 bgurejvfr; gura k_1 gb k_x unf vasvavgryl znal inyvq fhssvkrf sbe nal x.
Ner gurer frgf bs inyvq fgevatf fhpu gung ab Ghevat znpuvar N pbhyq vzcyrzrag guvf "nggnpx" ba gur bevtvany znpuvar G? Gung vf, qbrf gurer rkvfg n G fhpu gung, rira vs N unf npprff gb G'f fbhepr pbqr, N pnaabg gryy pbapyhfviryl juvpu bs K0 be K1 unf vasvavgryl znal inyvq fhssvkrf? Vs fb, pna lbh npghnyyl pbafgehpg n G gung cebinoyl unf guvf cebcregl?
ETA: Looks like cousin_it says that the answer to my existence question is "Yes."
The "original Turing machine" decides whether a prefix is valid? Then yes definitely; bear in mind Rice's theorem, which basically says that no non-trivial property of a Turing machine is computable.
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!