# Turing recognizable but not Turing decidable language cannot have TM do not halt on infinitely many inputs

Sorry, I think I misunderstand the question, It should read as if $$L$$ is turing-recognizable but not decidable, then there exists infinitely many input that any TM will not halt on it...

If $$L$$ is recognizable but not decidable, then $$L$$ has to be infinite (otherwise, it is decidable). If a TM recognizes such $$L$$, it has to accept all the words in $$L$$ (by definition of "recognizes"), hence it must halt in infinitely many cases.
• @Joe That machine will fail to halt on infinitely many inputs, otherwise (if it diverges only on finitely many cases), one could modify the TM on those finitely many cases so that it halts and rejects them and craft a TM that decides $L$, which is impossible. – chi Nov 12 '19 at 8:50