I learned that recursive language are decidable; correct me if I am wrong. However, I have found some arguments that seem to contradict this. These may or may not be correct; please let me know.
If a language is an REL (recursive enumerable language), I know that there exists a TM (Turing machine) that accepts it (regardless of the TM halting or not). Say, however, that for a language $L$ you have found a TM which accepts it, thus indicating $L$ is REL. We know that, given a TM, it is undecidable whether the TM halts or not. Thus, it is not possible to deduce whether $L$ is recursive or not: we have a TM that accepts $L$, but whether the TM halts or not is undecidable and, thus, we cannot comment on whether $L$ is recursive or not; this makes telling whether $L$ is recursive or not undecidable. Hence, recursive languages should be undecidable―which they are not!
What is wrong with the above reasoning?