The answer to your question is "It depends". Different axiomatic systems can lead to different conclusions. The truth of a statement S can only be said to be true or false within a given model.
This issue is not really a problem if you are talking about reaching a conclusion in two different axiomatic systems. We usually only work in one axiomatic system at a time and there is usually some kind of justification for using that system. And in this case, H1 and H2, in the OP's question would not be independent, as Conifold points out.
However, it is also possible for a single axiomatic system to lead to contradicting results. Such a system is said to be "inconsistent". We generally do not work with such systems and it is usually assumed that reality itself is logically consistent, but we do not have to make that assumption. There is a whole field of study involving inconsistent logical systems. They do have their benefits. For instance, in a consistent logical system, there are theorems which can never be proven true or false, while in an inconsistent logical system, we can always determine whether a theorem is true or false.
Additionally, Conifold's argument does not hold in a logically inconsistent system. Reductio ad absurdum, or proof by contradiction, is a form of valid proof, specifically because we have assumed that our logical system results in only true or false statements and that a statement cannot be both true and false.
The link I provided discusses this concept in more detail and also explains some reasons why we might not be certain that our reality is logically consistent and also how we can take a current axiomatic system and add to it in such a way as to make it useful but not consistent.