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to clarify the difference between this question and the supposed duplicate, absolutely nowhere do I mention the proof being attempted in the other question. It merely serves as an example related to this question.
Now, the way I view proof by contradiction is like this:
We have two logical statements $a$ and $b$ and we claim that "if $a$ then $b$".
We assume $b$ is false (given $a$) and we attempt to find some kind of contradiction with something we assume is true at the beginning (generally it tends to be proving $a$ is false).
Now, proof by contradiction can "fail" by resulting in no contradiction (i.e. that $a$ ends up being true when $b$ is false). However, this seems like a proof that the original statement is false. What I'm asking is more like how one detects or proves that when $b$ is false the truthfulness of $a$ is unprovable. Basically, that $a$ and $b$ are incapable of proving each other.
A good example of this is the parallel postulate of geometry. From what I heard (paraphrasing) "a failed proof by contradiction showed that there existed consistent systems where it was false and that it was unprovable from the other postulates". I'm not quite sure how that worked. Could someone please explain (in general) how one proves a proof by contradiction fails this brutally? I'm not asking specifically how it was done for the parallel postulate, but just how one goes about doing it in general?