While using the method of proof by contradiction, are we "assuming" consistency? I am aware of threads here and here which asks something similar. However, I had something very specific to ask under the same context.
I have a very elementary question about the connection between Godel's second incompleteness thorem and the method of proof by contradiction. From my limited knowledge that I have managed to gather from books and the internet, Godel's second theorem states that "No consistent, recursively
axiomatized theory that includes Peano Arithmetic can prove its own consistency." 
Consistency means that an axiomatic system cannot result in some statement, and its negation to be both simultaneously provable from the axioms or in other words, the axiomatic system is free of contradictions.
Proof by contradiction, I suppose, is based on the above fact. Since the axiomatic system is consistent, we should not be able to prove that a statement is both true and false at the same time. In other words, we are basically assuming that the system is free from contradictions based on which, we have proofs by contradiction. Proof by contradiction is routinely used in many mathematical disciplines like real analysis which I presume includes the Peano Arithmetic for the construction of real numbers.
However, Godel's second theorem states that we can never prove formally, within that system, its own consistency.
My question: Are we then just "assuming" consistency of the axiomatic system and then construct proofs based on contradiction? Consistency has to be assumed because Godel's second theorem makes it clear that we can never prove that the axiomatic system is consistent. Please clarify if consistency is assumed or if it is not so, please clarify the error in this thought process.
 A: We do not make that assumption, per se, when we prove things in a given system. Proof by contradiction is a valid rule of inference in (classical) logic... we can use it to prove things in a consistent system and we can use it to prove things in an inconsistent system. There is no assumption here... the rules are the rules.
Of course, if you are using an inconsistent set of axioms, you will be able to, in principle, to prove every statement. That's not a very good state of affairs from a philosophical standpoint.
So we want to know our axioms are consistent, not so that we can use proof by contradiction (again, that's cart-before-horse), but so that we know we're doing something meaningful. And Godel tells us (with a few caveats) that we'll never have a proof of a strong system from a weak (i.e. "safer") system, so tells us we will basically be assuming consistency rather than proving it in a philosophically satisfying way.
(Note also that even without knowledge of Godel's theorem, we would know that we need to take some consistency on faith since we need to assume the consistency of whatever simple "system" we make our most basic arguments about how proofs fit together. The hope that Godel destroys is that within this simple system, we might prove the consistency of a strong system like ZFC, where real heavy-duty mathematics is done)
