TITLE: Will assuming an undecidable statement result in a consistent system?
QUESTION [2 upvotes]: If you assume that an undecidable statement in a consistent axiomatic system is true (or false), will that new system also be consistent?
For example, does $\mathsf{ZF}$ being consistent imply that both $\mathsf{ZFC}$ and $\mathsf{ZF}\lnot \mathsf{C}$ are also consistent because the axiom of choice is undecidable in $\mathsf{ZF}$?
I believe that the answer is yes, because if assuming that it was true resulted in an inconsistent system (a contradiction), then it would have to be false, therefore not undecidable.

REPLY [2 votes]: I would use "disprovable," not "false," in your final sentence, but other than that, yes, you are correct. Saying "$\varphi$ is undecidable in $T$" is just another way of saying "Both $T\cup\{\varphi\}$ and $T\cup\{\neg\varphi\}$ are consistent," and this is due to the deduction theorem (https://en.wikipedia.org/wiki/Deduction_theorem). There are logics for which the deduction theorem fails, so this is not trivial (although there can be no complete logics for which the deduction theorem fails, for obvious reasons); see https://mathoverflow.net/questions/132268/deduction-theorem.
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