I am trying to see if infinite languages are always decidable. I believe it is not always decidable because there will not be a maximum length of string for the Turing machine to halt. Am I on the right path? I am having difficulty coming up with an example infinite language That is decidable.
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There are many infinite languages which are decidable, for example $\Sigma^*$, or the language of all (binary representations of) primes. However, in some sense "most" languages are undecidable, since there are uncountably many languages, but only countably many decidable languages. We also know many explicit undecidable languages, such as the language of (encodings of) Turing machines which halt on the empty input.
All finite languages, on the other hand, are decidable.
Yuval Filmus
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