Let's see: Is there a formal system to denote all complete systems and nothing else?
Completeness is not equal to deciding all recursive languages IMO. The bottom explanation of yours is not the same as the above question. Systems with sufficient power are incomplete. The system I am looking for doesn't have to decide its own termination, but its termination should be decideable by using an auxillary TM. Be careful with the jargon. Literature insists that "undecidable" is equal to "cannot be decided by a TM". Although an automaton may not decide its own halting , it may be decided by a TM, hence will be called decideable.
Let's try it in automata jargon:
We know that one needs a machine more powerful than an LBA to recognize all recursive languages. However, even an LBA, by definition is a TM and it can loop over two symbols forever. That is not all LBAs are guaranteed to halt.
Can there be an alternative machine formulation that only recognizes recursive sets and nothing else?
That is it should be stronger than an LBA, yet less powerful than a TM because it excludes R.E - recursive. So, we are looking for something that doesn't have a corresponding machine. It's possible
You mean "decide" not "recognize", right? that
it does not exist. But the proof could be interesting for us.
The fact that there may be looping LBAs is irrelevant because we can discard those, since the halting is decidable(again note the terminology, can be decided by a TM) for LBA. Indeed the system has to be powerful than an LBA but we still should be able to write a TM that is capabale of discarding the ones that go into infinite loop. I don't know whether it is correct to say that it should be more powerful than a LBA, because it won't have the power to loop :).