The diagonalization language, Ld, is a language that cannot be enumerated by any Turing machine. In other words, there is no Turing machine that can generate a list of all the strings in Ld.

To understand why Ld is recursively enumerable but not recursive, let's break down the definitions of these terms:

1. Recursive language:
- A language is recursive if there exists a Turing machine that can halt and accept or reject any given input string.
- This means that for any input string, the Turing machine will always reach a final state and provide a definite answer (accept or reject).

2. Recursive enumerable language:
- A language is recursively enumerable if there exists a Turing machine that can enumerate (list) all the strings in the language.
- This means that for any string in the language, the Turing machine will eventually generate it.
- However, the Turing machine may not halt or generate any output for strings that are not in the language.

Now, let's analyze Ld:

- Ld is the language that contains all the binary strings that do not belong to themselves. In other words, for any string x, x is in Ld if and only if x is not in the language described by the xth Turing machine.
- Assume that there exists a Turing machine TMd that can enumerate all the strings in Ld. This means that TMd can generate a list of all the binary strings that do not belong to themselves.
- Now, let's construct a contradiction by considering a new string d, which is the description of a Turing machine that behaves differently than all the Turing machines in the list generated by TMd.
- Since d is different from every Turing machine in the list, it must either belong to Ld or not. However, both possibilities lead to a contradiction:
1. If d is in Ld, it means that d does not belong to itself. But since d is different from all the machines in the list, it must belong to itself. This is a contradiction.
2. If d is not in Ld, it means that d belongs to itself. But again, since d is different from all the machines in the list, it must not belong to itself. This is also a contradiction.

The contradiction arises from assuming the existence of TMd, which means that no Turing machine can enumerate all the strings in Ld. Therefore, Ld is recursively enumerable but not recursive.