# Is the set of languages recognized by a Turing machine with an oracle countable?

I am currently taking a class in complexity theory and I am struggling with this question.

We define a TM with Oracle per. Sipser 6.18 as:

"An oracle for language $B$ is an external device that is capable of reporting whether any string $w$ is a member of $B$. An oracle Turing machine is a modified Turing machine that has the additional capability of querying an oracle. We write $M^b$ to describe an oracle Turing machine that has an oracle for language B."

I think that the answer is potentially false. We know that the set of languages is uncountable and suppose we had a Oracle Turing machine that accepts if the oracle answers YES, reject otherwise. Wouldn't that mean that then the set of languages recognized by TM with an oracle be uncountable?

• Yes, it depends on what the set of potential oracles is. – Raphael Nov 17 '17 at 6:38
• duplicate – Ariel Nov 17 '17 at 8:37