This seems related to these questions at a glance:

What are some problems which are easily solved by human brain but which would take more time computers?

What would show a human mind is/is not reducible to a Turing machine?

But not quite, I am not asking about "time", but power. Also, I am not interested in the turing test. That said, my question can also be expressed as two parts:

• Is there a language, which cannot be recognized by any turing machine, that can be recognized by a human?
• Is there a language, which cannot be decided by any turing machine, that can be decided by a human?

And vice versa. The "language" I am talking about is the "mathematical" language, not only a "human" or "programming" language:

$$L \subseteq \Sigma^*$$

Since this is a question about computational power, I would make the following assumptions:

• Human do not make mistakes
• There is no space limit (turing machine gets infinite tape, you get infinite medium to write)
• There are no time constraints
• However, the recognition/decision must be achieved within finite time.
• And of course, in finite space

Please give an example if you have an answer. Remember, this is a theoretical question, so practical issues are not in concern.