User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-39541634-20191030095026

There seems to be a typo in your first condition. Perhaps you meant the opposite? If so, then it is redundant, because it follows directly from conditions #2 and #3.

Condition #3 is part of the definition of what a "fundamental sequence" is, so it's not really needed.

Condition #4, however, seems to be the key. Or at least something like it. As written, I don't think your statement is well-defined (it mixes a couple of different things), but that should be the general idea.

Three possible additional obstacles:

(1) Regorously stating condition #4 might be tricker then it seems at first glance.

(2) Even if we manage (1), it may well be turn out to be a useless definition in practice because of uncomputability issues.

(3) We'll still need to prove that this condition (+condition 2) is actually sufficient. Just because it seems that way intuitively, does not make it true.