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.