是否存在所有可数语言的TM？

Question

I know that if a Turing machine exists for a language, that language is recursively enumerable and therefor there exists a enumeration procedure for it. However, if a language is countable, does that mean that there must be a TM for it?

Thanks!

Answer 1

The set Σ* is a countable set, so all its subsets are countable. This means, in particular, that every infinite language is countable, even though not all infinite languages are recursively enumerable. Therefore, there are infinitely many infinite languages for which no TM exists.

Hope this helps!