There are a countable number of TMs, since can encode all TMs as strings. But, it will be helpful and more satisfying to have a specific example. L such that for every recognizable language, can identify one reason why not equal. Actually, list all TM syntactic encodings in a canonical order. I.e., invert diagonal of table.
Diagonalization is a common proof method.
May have previously seen it to show that real numbers are uncountable. Intuitively, seems clear that an interpreter can't figure out whether its simulation won't halt, and to reject in that case. This is actually a diagonalization proof. Table's axes indexed by all TMs. Must be unrecognizable.