What is the difference between undecidable language and Turing Recognizable language?

cs.stackexchange https://cs.stackexchange.com/questions/88282

문제

I was wondering what is the difference difference undecidable language and Turing recognizable language. I've seen in some cases where they ask:

  1. Prove that the language $ A_{TM} = \{ \ <M,w> | \ M \mbox{ is a Turing Machine accepting } w\}$ is undecidable.
  2. Prove that the language $ A_{TM} = \{ \ <M,w> | \ M \mbox{ is a Turing Machine accepting } w\}$ is Turing Recognizable.

Are they the same thing? Can someone elaborate on this issue?

Another one, is the question at hand is known as the halting problem? Or is it different from halting problem?

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