It is high time the abuse of the Social Security Number ended. SSNs should be used for one thing: Social Security. Using a single "secret number" is an archaic system that for increasing numbers of people is no longer secret. Let's not forget all your other details which are used to identify you but aren't really that secret (your full name, your birthday, etc).
This information is used for identifying a person or proving identity so it's an authentication problem. We can do better! We have public key encryption. The government issues you a key pair (say, embedded into a photo ID, which we all have already) and now you can prove your identity without giving someone an irrevocable secret.
Authentication is also two factor: You have an ID and you know a PIN (or passphrase). If you lose your card, then your identity is not immediately compromised because it is protected by your PIN. This gives you time to have the gov't revoke your old key pair and issue you a new one.
In the case of the credit bureaus (I think we can all safely assume credit isn't going away any time soon), they associate your credit history with your public key and nothing else. If the key is revoked (by the gov't), then they move your file to the new key. No one can take out credit using the old key. In fact, any attempt could be reported to law enforcement.
The entire US Department of Defense has been using a system like this for years now and has by and large done away with things like passwords and hand signatures, especially for the things that matter most.
Is this completely foolproof to prevent someone impersonating you? No, but it is much better than having your SSN and other PII out on some forum where just anyone can use it for nefarious purposes and would be well worth its cost and complexity. The greatest obstacle is the credit bureaus having nothing to gain in actually protecting their "customers'" data because then to whom will they sell credit monitoring?